Theorem cbvexvw 1945

Description: Change bound variable. See cbvexv 1943 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1472. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvalvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexvw	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

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

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

Proof of Theorem cbvexvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	biimpd 144	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	equcoms 1732	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))
4	3	spimev 1885	. . 3 ⊢ (𝜑 → ∃𝑦𝜓)
5	4	exlimiv 1622	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
6	1	biimprd 158	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	6	spimev 1885	. . 3 ⊢ (𝜓 → ∃𝑥𝜑)
8	7	exlimiv 1622	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
9	5, 8	impbii 126	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485

This theorem is referenced by:  cbvex2vw  1958  cbvrexvw  2744  prodmodc  11959