Theorem bj-pm11.53v 37769

Description: Version of pm11.53v 1976 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)

Assertion

Ref	Expression
bj-pm11.53v	⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-pm11.53v

Step	Hyp	Ref	Expression
1		bj-nnfalt 37767	. 2 ⊢ (∀𝑦Ⅎ'𝑥𝜓 → Ⅎ'𝑥∀𝑦𝜓)
2		bj-pm11.53vw 37744	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
3	1, 2	sylan2 605	1 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  Ⅎ'wnnf 37703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-bj-nnf 37704

This theorem is used by: (None)