Theorem bj-pm11.53vw 34644

Description: Version of pm11.53v 1952 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)

Assertion

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

Proof of Theorem bj-pm11.53vw

Step	Hyp	Ref	Expression
1		simpl 486	. . . 4 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → ∀𝑥Ⅎ'𝑦𝜑)
2		bj-19.21t 34637	. . . 4 ⊢ (Ⅎ'𝑦𝜑 → (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
3	1, 2	sylg 1830	. . 3 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → ∀𝑥(∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
4		albi 1826	. . 3 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)))
5	3, 4	syl 17	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)))
6		bj-19.23t 34638	. . 3 ⊢ (Ⅎ'𝑥∀𝑦𝜓 → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
7	6	adantl 485	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
8	5, 7	bitrd 282	1 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787  Ⅎ'wnnf 34591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-bj-nnf 34592

This theorem is referenced by:  bj-pm11.53v  34645  bj-pm11.53a  34646