Theorem gencbvex 3513

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
gencbvex.1	⊢ 𝐴 ∈ V
gencbvex.2	⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
gencbvex.3	⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
gencbvex.4	⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))

Assertion

Ref	Expression
gencbvex	⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜃,𝑥   𝜒,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐴(𝑥)

Proof of Theorem gencbvex

Step	Hyp	Ref	Expression
1		excom 2199	. 2 ⊢ (∃𝑥∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑦∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
2		gencbvex.1	. . . 4 ⊢ 𝐴 ∈ V
3		gencbvex.3	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
4		gencbvex.2	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	anbi12d 643	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝜒 ∧ 𝜑) ↔ (𝜃 ∧ 𝜓)))
6	5	bicomd 226	. . . . 5 ⊢ (𝐴 = 𝑦 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
7	6	eqcoms 2773	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
8	2, 7	ceqsexv 3505	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜒 ∧ 𝜑))
9	8	exbii 1871	. 2 ⊢ (∃𝑥∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑥(𝜒 ∧ 𝜑))
10		19.41v 1972	. . . 4 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
11		simpr 489	. . . . 5 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) → (𝜃 ∧ 𝜓))
12		gencbvex.4	. . . . . . . 8 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
13		eqcom 2772	. . . . . . . . . 10 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
14	13	bilani 509	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝑦 = 𝐴)
15	14	eximi 1858	. . . . . . . 8 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
16	12, 15	sylbi 220	. . . . . . 7 ⊢ (𝜃 → ∃𝑥 𝑦 = 𝐴)
17	16	adantr 485	. . . . . 6 ⊢ ((𝜃 ∧ 𝜓) → ∃𝑥 𝑦 = 𝐴)
18	17	ancri 558	. . . . 5 ⊢ ((𝜃 ∧ 𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
19	11, 18	impbii 212	. . . 4 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
20	10, 19	bitri 278	. . 3 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
21	20	exbii 1871	. 2 ⊢ (∃𝑦∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑦(𝜃 ∧ 𝜓))
22	1, 9, 21	3bitr3i 304	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-cleq 2757  df-clel 2840

This theorem is referenced by:  gencbvex2  3514  gencbval  3515