Theorem bj-cbvexw 34011

Description: Change bound variable. This is to cbvexvw 2044 what cbvalw 2042 is to cbvalvw 2043. (Contributed by BJ, 17-Mar-2020.)

Hypotheses

Ref	Expression
bj-cbvexw.1	⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
bj-cbvexw.2	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbvexw.3	⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)
bj-cbvexw.4	⊢ (𝜓 → ∀𝑥𝜓)
bj-cbvexw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-cbvexw	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-cbvexw

Step	Hyp	Ref	Expression
1		bj-cbvexw.1	. . 3 ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
2		bj-cbvexw.2	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbvexw.5	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	equcoms 2027	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
5	4	biimpd 231	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))
6	1, 2, 5	bj-cbvexiw 34006	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
7		bj-cbvexw.3	. . 3 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)
8		bj-cbvexw.4	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
9	3	biimprd 250	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
10	7, 8, 9	bj-cbvexiw 34006	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
11	6, 10	impbii 211	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)