Theorem cgsex4g 2800

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

Hypotheses

Ref	Expression
cgsex4g.1	⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
cgsex4g.2	⊢ (𝜒 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cgsex4g	⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝜓,𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜒(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦,𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cgsex4g

Step	Hyp	Ref	Expression
1		cgsex4g.2	. . . . 5 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 296	. . . 4 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1911	. . 3 ⊢ (∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
4	3	exlimivv 1911	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
5		elisset 2777	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑅 → ∃𝑥 𝑥 = 𝐴)
6		elisset 2777	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑆 → ∃𝑦 𝑦 = 𝐵)
7	5, 6	anim12i 338	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		eeanv 1951	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9	7, 8	sylibr 134	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		elisset 2777	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑅 → ∃𝑧 𝑧 = 𝐶)
11		elisset 2777	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑆 → ∃𝑤 𝑤 = 𝐷)
12	10, 11	anim12i 338	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13		eeanv 1951	. . . . . . 7 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
14	12, 13	sylibr 134	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
15	9, 14	anim12i 338	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		ee4anv 1953	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
17	15, 16	sylibr 134	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
18		cgsex4g.1	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
19	18	2eximi 1615	. . . . 5 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑧∃𝑤𝜒)
20	19	2eximi 1615	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
21	17, 20	syl 14	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
22	1	biimprcd 160	. . . . . 6 ⊢ (𝜓 → (𝜒 → 𝜑))
23	22	ancld 325	. . . . 5 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
24	23	2eximdv 1896	. . . 4 ⊢ (𝜓 → (∃𝑧∃𝑤𝜒 → ∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
25	24	2eximdv 1896	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜒 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
26	21, 25	syl5com 29	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (𝜓 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
27	4, 26	impbid2 143	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  copsex4g  4280  brecop  6684