Theorem cgsex4gOLD 3529

Description: Obsolete version of cgsex4g 3528 as of 21-Mar-2025. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2141, ax-11 2157. (Revised by GG, 28-Jun-2024.) (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

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgsex4gOLD.2	. . . . 5 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 476	. . . 4 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1932	. . 3 ⊢ (∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
4	3	exlimivv 1932	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
5		elisset 2823	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑅 → ∃𝑥 𝑥 = 𝐴)
6		elisset 2823	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑆 → ∃𝑦 𝑦 = 𝐵)
7	5, 6	anim12i 613	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		exdistrv 1955	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9	7, 8	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		elisset 2823	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑅 → ∃𝑧 𝑧 = 𝐶)
11		elisset 2823	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑆 → ∃𝑤 𝑤 = 𝐷)
12	10, 11	anim12i 613	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13		exdistrv 1955	. . . . . . 7 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
14	12, 13	sylibr 234	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
15	9, 14	anim12i 613	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝑦 = 𝐵 ↔ 𝑣 = 𝐵))
17	16	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑣 = 𝐵)))
18	17	anbi1d 631	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
19	18	exbidv 1921	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
20	19	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
21		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑣 → (𝑧 = 𝐶 ↔ 𝑣 = 𝐶))
22	21	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷)))
23	22	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
24	23	exbidv 1921	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
25	24	notbid 318	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
26	20, 25	alcomw 2044	. . . . . . . . 9 ⊢ (∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
27	26	notbii 320	. . . . . . . 8 ⊢ (¬ ∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
28		2exnaln 1829	. . . . . . . 8 ⊢ (∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
29		2exnaln 1829	. . . . . . . 8 ⊢ (∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
30	27, 28, 29	3bitr4i 303	. . . . . . 7 ⊢ (∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
31	30	exbii 1848	. . . . . 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)))
32		4exdistrv 1956	. . . . . 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)))
33	31, 32	bitri 275	. . . . 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)))
34	15, 33	sylibr 234	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
35		cgsex4gOLD.1	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
36	35	2eximi 1836	. . . . 5 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑧∃𝑤𝜒)
37	36	2eximi 1836	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
38	34, 37	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
39	1	biimprcd 250	. . . . . 6 ⊢ (𝜓 → (𝜒 → 𝜑))
40	39	ancld 550	. . . . 5 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
41	40	2eximdv 1919	. . . 4 ⊢ (𝜓 → (∃𝑧∃𝑤𝜒 → ∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
42	41	2eximdv 1919	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜒 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
43	38, 42	syl5com 31	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (𝜓 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
44	4, 43	impbid2 226	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816

This theorem is referenced by: (None)