Theorem bj-pm11.53v 35655

Description: Version of pm11.53v 1948 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 35644	. 2 ⊢ (∀𝑦Ⅎ'𝑥𝜓 → Ⅎ'𝑥∀𝑦𝜓)
2		bj-pm11.53vw 35654	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
3	1, 2	sylan2 594	1 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540  ∃wex 1782  Ⅎ'wnnf 35601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-bj-nnf 35602

This theorem is referenced by: (None)