Theorem cgsex2g 3465

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Hypotheses

Ref	Expression
cgsex2g.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒)
cgsex2g.2	⊢ (𝜒 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cgsex2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . 4 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 476	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1936	. 2 ⊢ (∃𝑥∃𝑦(𝜒 ∧ 𝜑) → 𝜓)
4		elisset 2820	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		elisset 2820	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
6	4, 5	anim12i 612	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7		exdistrv 1960	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8	6, 7	sylibr 233	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9		cgsex2g.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒)
10	9	2eximi 1839	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜒)
11	8, 10	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦𝜒)
12	1	biimprcd 249	. . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑))
13	12	ancld 550	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
14	13	2eximdv 1923	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦𝜒 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑)))
15	11, 14	syl5com 31	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑)))
16	3, 15	impbid2 225	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-clel 2817

This theorem is referenced by:  copsex2g  5401