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Proof step 37 from proof del-exists (previous step)

st0 = st[a, st0[a]], a ∈ st0, ∀ a. a ∈ st0 → (∃ st1. ¬ a ∈ st1 ∧ st0 = st1[a, st0[a]]) ⊦ a = a0, st0[a0, d] = st[a0, d][a, st0[a]], a ∈ st, ∃ st1. ¬ a ∈ st1 ∧ st0[a0, d] = st1[a, st0[a]]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ st1 = st2 ↔ (∀ a. (a ∈ st1 ↔ a ∈ st2) ∧ st1[a] = st2[a])
Proof step 39:
st0 = st[a, st0[a]], a ∈ st0, ∀ a. a ∈ st0 → (∃ st1. ¬ a ∈ st1 ∧ st0 = st1[a, st0[a]]) ⊦ a = a0, ∀ a1. (a1 ∈ st0[a0, d] ↔ a1 ∈ st[a0, d][a, st0[a]]) ∧ st0[a0, d][a1] = st[a0, d][a, st0[a]][a1], a ∈ st, ∃ st1. ¬ a ∈ st1 ∧ st0[a0, d] = st1[a, st0[a]]

Rule argument

spec: inst: name: Extension seq: ⊦ st1 = st2 ↔ (∀ a. (a ∈ st1 ↔ a ∈ st2) ∧ st1[a] = st2[a]) subst: [st2, st1] → [st[a0, d][a, st0[a]], st0[a0, d]] flags: rewrite, current

Additional information

Used terms: st[a0, d][a, st0[a]], st0[a0, d]