Theorem cgsex4g 3466

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2139, ax-11 2156. (Revised by Gino Giotto, 28-Jun-2024.)

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

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgsex4g.2	. . . . 5 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
2	1	biimpa 476	. . . 4 ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
3	2	exlimivv 1936	. . 3 ⊢ (∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
4	3	exlimivv 1936	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) → 𝜓)
5		elisset 2820	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑅 → ∃𝑥 𝑥 = 𝐴)
6		elisset 2820	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑆 → ∃𝑦 𝑦 = 𝐵)
7	5, 6	anim12i 612	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		exdistrv 1960	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9	7, 8	sylibr 233	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
10		elisset 2820	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑅 → ∃𝑧 𝑧 = 𝐶)
11		elisset 2820	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑆 → ∃𝑤 𝑤 = 𝐷)
12	10, 11	anim12i 612	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
13		exdistrv 1960	. . . . . . 7 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (∃𝑧 𝑧 = 𝐶 ∧ ∃𝑤 𝑤 = 𝐷))
14	12, 13	sylibr 233	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷))
15	9, 14	anim12i 612	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
16		eqeq1 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑣 → (𝑧 = 𝐶 ↔ 𝑣 = 𝐶))
17	16	anbi1d 629	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑣 → ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ↔ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷)))
18	17	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
19	18	exbidv 1925	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
20	19	notbid 317	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑤 = 𝐷))))
21	20	alcomiw 2047	. . . . . . . . . 10 ⊢ (∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
22		eqeq1 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑣 → (𝑦 = 𝐵 ↔ 𝑣 = 𝐵))
23	22	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑣 = 𝐵)))
24	23	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
25	24	exbidv 1925	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑤((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
26	25	notbid 317	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷))))
27	26	alcomiw 2047	. . . . . . . . . 10 ⊢ (∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
28	21, 27	impbii 208	. . . . . . . . 9 ⊢ (∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
29	28	notbii 319	. . . . . . . 8 ⊢ (¬ ∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
30		2exnaln 1832	. . . . . . . 8 ⊢ (∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑦∀𝑧 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
31		2exnaln 1832	. . . . . . . 8 ⊢ (∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ¬ ∀𝑧∀𝑦 ¬ ∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
32	29, 30, 31	3bitr4i 302	. . . . . . 7 ⊢ (∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ↔ ∃𝑧∃𝑦∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
33	32	exbii 1851	. . . . . 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)))
34		4exdistrv 1961	. . . . . 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)))
35	33, 34	bitri 274	. . . . 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)))
36	15, 35	sylibr 233	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)))
37		cgsex4g.1	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)
38	37	2eximi 1839	. . . . 5 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑧∃𝑤𝜒)
39	38	2eximi 1839	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
40	36, 39	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)
41	1	biimprcd 249	. . . . . 6 ⊢ (𝜓 → (𝜒 → 𝜑))
42	41	ancld 550	. . . . 5 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑)))
43	42	2eximdv 1923	. . . 4 ⊢ (𝜓 → (∃𝑧∃𝑤𝜒 → ∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
44	43	2eximdv 1923	. . 3 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜒 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
45	40, 44	syl5com 31	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (𝜓 → ∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑)))
46	4, 45	impbid2 225	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817

This theorem is referenced by:  copsex4g  5403  brecop  8557