Theorem cgsex2g 3499

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 481	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1961	. 2 ⊢ (∃𝑥∃𝑦(𝜒 ∧ 𝜑) → 𝜓)
4		elisset 2844	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		elisset 2844	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
6	4, 5	anim12i 624	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
7		exdistrv 1984	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8	6, 7	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
9		cgsex2g.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒)
10	9	2eximi 1865	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜒)
11	8, 10	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦𝜒)
12	1	biimprcd 253	. . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑))
13	12	ancld 559	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
14	13	2eximdv 1948	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦𝜒 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑)))
15	11, 14	syl5com 32	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑)))
16	3, 15	impbid2 229	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-clel 2837

This theorem is used by:  copsex2g  5475