Theorem bj-cbveximi 34756

Description: An equality-free general instance of one half of a precise form of bj-cbvex 34758. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbvalimi.maj	⊢ (𝜒 → (𝜑 → 𝜓))
bj-cbveximi.denote	⊢ ∀𝑥∃𝑦𝜒

Assertion

Ref	Expression
bj-cbveximi	⊢ (∃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-cbveximi

Step	Hyp	Ref	Expression
1		bj-cbveximi.denote	. 2 ⊢ ∀𝑥∃𝑦𝜒
2		bj-cbvalimi.maj	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
3	2	gen2 1800	. 2 ⊢ ∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓))
4		bj-cbvexim 34754	. 2 ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))
5	1, 3, 4	mp2 9	1 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  bj-cbvex  34758