Theorem bj-pm11.53a 37427

Description: A variant of pm11.53v 1977. 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 37425	. 2 ⊢ Ⅎ'𝑥∀𝑦𝜓
2		bj-pm11.53vw 37424	. 2 ⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
3	1, 2	mpan2 704	1 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37384

This theorem is used by: (None)