Theorem gencbvex 2600

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 1554	. 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 442	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝜒 ∧ 𝜑) ↔ (𝜃 ∧ 𝜓)))
6	5	bicomd 129	. . . . 5 ⊢ (𝐴 = 𝑦 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
7	6	eqcoms 2043	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜑)))
8	2, 7	ceqsexv 2593	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜒 ∧ 𝜑))
9	8	exbii 1496	. 2 ⊢ (∃𝑥∃𝑦(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑥(𝜒 ∧ 𝜑))
10		19.41v 1782	. . . 4 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
11		simpr 103	. . . . 5 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) → (𝜃 ∧ 𝜓))
12		gencbvex.4	. . . . . . . 8 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
13		eqcom 2042	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
14	13	biimpi 113	. . . . . . . . . 10 ⊢ (𝐴 = 𝑦 → 𝑦 = 𝐴)
15	14	adantl 262	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝑦 = 𝐴)
16	15	eximi 1491	. . . . . . . 8 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
17	12, 16	sylbi 114	. . . . . . 7 ⊢ (𝜃 → ∃𝑥 𝑦 = 𝐴)
18	17	adantr 261	. . . . . 6 ⊢ ((𝜃 ∧ 𝜓) → ∃𝑥 𝑦 = 𝐴)
19	18	ancri 307	. . . . 5 ⊢ ((𝜃 ∧ 𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)))
20	11, 19	impbii 117	. . . 4 ⊢ ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
21	10, 20	bitri 173	. . 3 ⊢ (∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ (𝜃 ∧ 𝜓))
22	21	exbii 1496	. 2 ⊢ (∃𝑦∃𝑥(𝑦 = 𝐴 ∧ (𝜃 ∧ 𝜓)) ↔ ∃𝑦(𝜃 ∧ 𝜓))
23	1, 9, 22	3bitr3i 199	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  gencbvex2  2601