Theorem cgsex4g 3213

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 500	. . . 4 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1847	. . 3 ⊢ (∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
4	3	exlimivv 1847	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
5		elisset 3188	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑅 → ∃𝑥 𝑥 = 𝐴)
6		elisset 3188	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑆 → ∃𝑦 𝑦 = 𝐵)
7	5, 6	anim12i 588	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		eeanv 2170	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9	7, 8	sylibr 223	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		elisset 3188	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑅 → ∃𝑧 𝑧 = 𝐶)
11		elisset 3188	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑆 → ∃𝑤 𝑤 = 𝐷)
12	10, 11	anim12i 588	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13		eeanv 2170	. . . . . . 7 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
14	12, 13	sylibr 223	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
15	9, 14	anim12i 588	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		ee4anv 2172	. . . . 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 223	. . . 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 1753	. . . . 5 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑧∃𝑤𝜒)
20	19	2eximi 1753	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
21	17, 20	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
22	1	biimprcd 239	. . . . . 6 ⊢ (𝜓 → (𝜒 → 𝜑))
23	22	ancld 574	. . . . 5 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
24	23	2eximdv 1835	. . . 4 ⊢ (𝜓 → (∃𝑧∃𝑤𝜒 → ∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
25	24	2eximdv 1835	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜒 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
26	21, 25	syl5com 31	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (𝜓 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
27	4, 26	impbid2 215	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  copsex4g  4885  brecop  7727