Theorem bj-pm11.53vw 36752

Description: Version of pm11.53v 1943 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 482	. . . 4 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → ∀𝑥Ⅎ'𝑦𝜑)
2		bj-19.21t 36745	. . . 4 ⊢ (Ⅎ'𝑦𝜑 → (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
3	1, 2	sylg 1822	. . 3 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → ∀𝑥(∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
4		albi 1817	. . 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 36746	. . 3 ⊢ (Ⅎ'𝑥∀𝑦𝜓 → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
7	6	adantl 481	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
8	5, 7	bitrd 279	1 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778  Ⅎ'wnnf 36699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-bj-nnf 36700

This theorem is referenced by:  bj-pm11.53v  36753  bj-pm11.53a  36754