Theorem bj-pm11.53a 36757

Description: A variant of pm11.53v 1944. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.)

Assertion

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

Distinct variable groups:   𝜓,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-pm11.53a

Step	Hyp	Ref	Expression
1		bj-nnfv 36733	. 2 ⊢ Ⅎ'𝑥∀𝑦𝜓
2		bj-pm11.53vw 36755	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
3	1, 2	mpan2 691	1 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779  Ⅎ'wnnf 36702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-bj-nnf 36703

This theorem is referenced by: (None)