Theorem bj-cbvexiw 34589

Description: Change bound variable. This is to cbvexvw 2045 what cbvaliw 2014 is to cbvalvw 2044. TODO: move after cbvalivw 2015. (Contributed by BJ, 17-Mar-2020.)

Hypotheses

Ref	Expression
bj-cbvexiw.1	⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
bj-cbvexiw.2	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbvexiw.3	⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-cbvexiw	⊢ (∃𝑥𝜑 → ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-cbvexiw

Step	Hyp	Ref	Expression
1		bj-cbvexiw.1	. 2 ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
2		bj-cbvexiw.2	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbvexiw.3	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))
4	2, 3	spimew 1980	. 2 ⊢ (𝜑 → ∃𝑦𝜓)
5	1, 4	bj-sylge 34542	1 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-6 1976

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  bj-cbvexivw  34590  bj-cbvexw  34594