Theorem cgsexg 2634

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Hypotheses

Ref	Expression
cgsexg.1	⊢ (𝑥 = 𝐴 → 𝜒)
cgsexg.2	⊢ (𝜒 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cgsexg	⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝑉(𝑥)

Proof of Theorem cgsexg

Step	Hyp	Ref	Expression
1		cgsexg.2	. . . 4 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 290	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimiv 1529	. 2 ⊢ (∃𝑥(𝜒 ∧ 𝜑) → 𝜓)
4		elisset 2613	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		cgsexg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝜒)
6	5	eximi 1531	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
7	4, 6	syl 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥𝜒)
8	1	biimprcd 158	. . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑))
9	8	ancld 318	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
10	9	eximdv 1801	. . 3 ⊢ (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒 ∧ 𝜑)))
11	7, 10	syl5com 29	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥(𝜒 ∧ 𝜑)))
12	3, 11	impbid2 141	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433

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-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)