Theorem cgsexg 3238

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 501	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimiv 1858	. 2 ⊢ (∃𝑥(𝜒 ∧ 𝜑) → 𝜓)
4		elisset 3215	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		cgsexg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝜒)
6	5	eximi 1762	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
7	4, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥𝜒)
8	1	biimprcd 240	. . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑))
9	8	ancld 576	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
10	9	eximdv 1846	. . 3 ⊢ (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒 ∧ 𝜑)))
11	7, 10	syl5com 31	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥(𝜒 ∧ 𝜑)))
12	3, 11	impbid2 216	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)