Theorem cbvexh 1678

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 1567	. . 3 ⊢ (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)
3		cbvexh.1	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
4		cbvexh.3	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	bicomd 139	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6	5	equcoms 1634	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	3, 6	equsex 1656	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) ↔ 𝜑)
8		simpr 108	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝜓) → 𝜓)
9	8	eximi 1531	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜓) → ∃𝑦𝜓)
10	7, 9	sylbir 133	. . 3 ⊢ (𝜑 → ∃𝑦𝜓)
11	2, 10	exlimih 1524	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
12	3	hbex 1567	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)
13	1, 4	equsex 1656	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
14		simpr 108	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)
15	14	eximi 1531	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
16	13, 15	sylbir 133	. . 3 ⊢ (𝜓 → ∃𝑥𝜑)
17	12, 16	exlimih 1524	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
18	11, 17	impbii 124	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cbvex  1679  sb8eh  1776  cbvexv  1836  euf  1946  mopick  2019