Theorem cbvexv1 1766

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 1770 with a disjoint variable condition. See cbvexvw 1935 for a version with two disjoint variable conditions, and cbvexv 1933 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	⊢ Ⅎ𝑦𝜑
cbvalv1.nf2	⊢ Ⅎ𝑥𝜓
cbvalv1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexv1	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf2	. . . 4 ⊢ Ⅎ𝑥𝜓
2	1	nfex 1651	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
3		cbvalv1.nf1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
4	3	nfri 1533	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
5		cbvalv1.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	bicomd 141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
7	6	equcoms 1722	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
8	4, 7	equsex 1742	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) ↔ 𝜑)
9		exsimpr 1632	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) → ∃𝑦𝜓)
10	8, 9	sylbir 135	. . 3 ⊢ (𝜑 → ∃𝑦𝜓)
11	2, 10	exlimi 1608	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
12	3	nfex 1651	. . 3 ⊢ Ⅎ𝑦∃𝑥𝜑
13	1	nfri 1533	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
14	13, 5	equsex 1742	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
15		exsimpr 1632	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
16	14, 15	sylbir 135	. . 3 ⊢ (𝜓 → ∃𝑥𝜑)
17	12, 16	exlimi 1608	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
18	11, 17	impbii 126	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  Ⅎwnf 1474  ∃wex 1506

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  cbvrexfw  2720  fprod2dlemstep  11787