Theorem bj-cbvalimi 33975

Description: An equality-free general instance of one half of a precise form of bj-cbval 33977. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbvalimi.maj	⊢ (𝜒 → (𝜑 → 𝜓))
bj-cbvalimi.denote	⊢ ∀𝑦∃𝑥𝜒

Assertion

Ref	Expression
bj-cbvalimi	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

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

Proof of Theorem bj-cbvalimi

Step	Hyp	Ref	Expression
1		bj-cbvalimi.denote	. 2 ⊢ ∀𝑦∃𝑥𝜒
2		bj-cbvalimi.maj	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
3	2	gen2 1793	. 2 ⊢ ∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓))
4		bj-cbvalim 33973	. 2 ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
5	1, 3, 4	mp2 9	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  bj-cbval  33977