Theorem bj-cbvexivw 31847

Description: Change bound variable. This is to cbvexvw 1957 what cbvalivw 1921 is to cbvalvw 1956. [TODO: move after cbvalivw 1921]. (Contributed by BJ, 17-Mar-2020.)

Hypothesis

Ref	Expression
bj-cbvexivw.1	⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))

Assertion

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

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

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

Proof of Theorem bj-cbvexivw

Step	Hyp	Ref	Expression
1		ax5e 1829	. 2 ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
2		ax-5 1827	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbvexivw.1	. 2 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))
4	1, 2, 3	bj-cbvexiw 31846	1 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)