Theorem gencbvex 3540

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 2161	. 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 632	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝜒 ∧ 𝜑) ↔ (𝜃 ∧ 𝜓)))
6	5	bicomd 223	. . . . 5 ⊢ (𝐴 = 𝑦 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
7	6	eqcoms 2744	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
8	2, 7	ceqsexv 3531	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜒 ∧ 𝜑))
9	8	exbii 1847	. 2 ⊢ (∃𝑥∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑥(𝜒 ∧ 𝜑))
10		19.41v 1948	. . . 4 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
11		simpr 484	. . . . 5 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) → (𝜃 ∧ 𝜓))
12		gencbvex.4	. . . . . . . 8 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
13		eqcom 2743	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
14	13	biimpi 216	. . . . . . . . . 10 ⊢ (𝐴 = 𝑦 → 𝑦 = 𝐴)
15	14	adantl 481	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝑦 = 𝐴)
16	15	eximi 1834	. . . . . . . 8 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
17	12, 16	sylbi 217	. . . . . . 7 ⊢ (𝜃 → ∃𝑥 𝑦 = 𝐴)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜃 ∧ 𝜓) → ∃𝑥 𝑦 = 𝐴)
19	18	ancri 549	. . . . 5 ⊢ ((𝜃 ∧ 𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
20	11, 19	impbii 209	. . . 4 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
21	10, 20	bitri 275	. . 3 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
22	21	exbii 1847	. 2 ⊢ (∃𝑦∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑦(𝜃 ∧ 𝜓))
23	1, 9, 22	3bitr3i 301	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728  df-clel 2815

This theorem is referenced by:  gencbvex2  3541  gencbval  3542