Theorem cbvexh 1769

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Hypotheses

Ref	Expression
cbvexh.1	⊢ (𝜑 → ∀𝑦𝜑)
cbvexh.2	⊢ (𝜓 → ∀𝑥𝜓)
cbvexh.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexh	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Proof of Theorem cbvexh

Step	Hyp	Ref	Expression
1		cbvexh.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	hbex 1650	. . 3 ⊢ (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)
3		cbvexh.1	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
4		cbvexh.3	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	bicomd 141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6	5	equcoms 1722	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	3, 6	equsex 1742	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) ↔ 𝜑)
8		simpr 110	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝜓) → 𝜓)
9	8	eximi 1614	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) → ∃𝑦𝜓)
10	7, 9	sylbir 135	. . 3 ⊢ (𝜑 → ∃𝑦𝜓)
11	2, 10	exlimih 1607	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
12	3	hbex 1650	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)
13	1, 4	equsex 1742	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
14		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)
15	14	eximi 1614	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
16	13, 15	sylbir 135	. . 3 ⊢ (𝜓 → ∃𝑥𝜑)
17	12, 16	exlimih 1607	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
18	11, 17	impbii 126	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃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

This theorem is referenced by:  cbvex  1770  sb8eh  1869  cbvexv  1933  euf  2050  mopick  2123