Theorem cgsexg 3484

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 480	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimiv 1931	. 2 ⊢ (∃𝑥(𝜒 ∧ 𝜑) → 𝜓)
4		elisset 3452	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		cgsexg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝜒)
6	5	eximi 1836	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
7	4, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥𝜒)
8	1	biimprcd 253	. . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑))
9	8	ancld 554	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
10	9	eximdv 1918	. . 3 ⊢ (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒 ∧ 𝜑)))
11	7, 10	syl5com 31	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥(𝜒 ∧ 𝜑)))
12	3, 11	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870

This theorem is referenced by:  ceqsexgv  3595