Theorem cgsex4gOLD 3455

Description: Obsolete version of cgsex4g 3454 as of 28-Jun-2024. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2143. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cgsex4gOLD.1	⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
cgsex4gOLD.2	⊢ (𝜒 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cgsex4gOLD	⊢ (((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 cgsex4gOLD

Step	Hyp	Ref	Expression
1		cgsex4gOLD.2	. . . . 5 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 481	. . . 4 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1934	. . 3 ⊢ (∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
4	3	exlimivv 1934	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
5		elisset 3420	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑅 → ∃𝑥 𝑥 = 𝐴)
6		elisset 3420	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑆 → ∃𝑦 𝑦 = 𝐵)
7	5, 6	anim12i 616	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		exdistrv 1957	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9	7, 8	sylibr 237	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		elisset 3420	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑅 → ∃𝑧 𝑧 = 𝐶)
11		elisset 3420	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑆 → ∃𝑤 𝑤 = 𝐷)
12	10, 11	anim12i 616	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13		exdistrv 1957	. . . . . . 7 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
14	12, 13	sylibr 237	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
15	9, 14	anim12i 616	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		excom 2167	. . . . . . 7 ⊢ (∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
17	16	exbii 1850	. . . . . 6 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
18		4exdistrv 1958	. . . . . 6 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
19	17, 18	bitri 278	. . . . 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)))
20	15, 19	sylibr 237	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
21		cgsex4gOLD.1	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
22	21	2eximi 1838	. . . . 5 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑧∃𝑤𝜒)
23	22	2eximi 1838	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
24	20, 23	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
25	1	biimprcd 253	. . . . . 6 ⊢ (𝜓 → (𝜒 → 𝜑))
26	25	ancld 555	. . . . 5 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
27	26	2eximdv 1921	. . . 4 ⊢ (𝜓 → (∃𝑧∃𝑤𝜒 → ∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
28	27	2eximdv 1921	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜒 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
29	24, 28	syl5com 31	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (𝜓 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
30	4, 29	impbid2 229	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539  ∃wex 1782   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-cleq 2751  df-clel 2831

This theorem is referenced by: (None)