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Proof step 21 from proof cont-monotone (previous step)

f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x) ↔ (∃ n. f((n = 0 ⊃ S0;S))(x)), f(S0)(x), S0 ⊆ S ⊦ f(S)(x)

User applied rule case distinction

And obtained the following new goals

Proof step 22:
f(S0)(x), S0 ⊆ S ⊦ f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x), ∃ n. f((n = 0 ⊃ S0;S))(x), f(S)(x)
Proof step 23:
f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x), ∃ n. f((n = 0 ⊃ S0;S))(x), f(S0)(x), S0 ⊆ S ⊦ f(S)(x)

Rule argument

pos: left 1

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