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Proof step 7 from proof keys-in-elim (previous step)

∀ key, fs. key ∈ keys(fs, ax0) ↔ (∃ adr. adr ∈ ax0 ∧ fs[adr].key = key) ⊦ key ∈ keys(fs, ax0) ↔ (∃ adr0. adr0 ∈ ax0 ∧ fs[adr0].key = key), key = fs[adr].key

Heuristic cut applied rule cut formula

And obtained the following new goals

Proof step 8:
key ∈ keys(fs, ax0), ∀ key, fs. key ∈ keys(fs, ax0) ↔ (∃ adr. adr ∈ ax0 ∧ fs[adr].key = key) ⊦ key ∈ keys(fs, ax0) ↔ (∃ adr0. adr0 ∈ ax0 ∧ fs[adr0].key = key), key = fs[adr].key
Proof step 12:
∀ key, fs. key ∈ keys(fs, ax0) ↔ (∃ adr. adr ∈ ax0 ∧ fs[adr].key = key) ⊦ key ∈ keys(fs, ax0) ↔ (∃ adr0. adr0 ∈ ax0 ∧ fs[adr0].key = key), key = fs[adr].key, key ∈ keys(fs, ax0)

Rule argument

fma: key ∈ keys(fs, ax0)

Additional information

Used formulas: key ∈ keys(fs, ax0)