Theorem bj-pm11.53v 36735

Description: Version of pm11.53v 1943 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 36724	. 2 ⊢ (∀𝑦Ⅎ'𝑥𝜓 → Ⅎ'𝑥∀𝑦𝜓)
2		bj-pm11.53vw 36734	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
3	1, 2	sylan2 592	1 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  Ⅎ'wnnf 36681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-bj-nnf 36682

This theorem is referenced by: (None)