Theorem cgsex4g 1369

Description: An implicit substitution inference for 4 general classes.

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 group(s):   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

Proof of Theorem cgsex4g

Step	Hyp	Ref	Expression
1		cgsex4g.2	. . . . . 6 ⊢ (χ → (φ ↔ ψ))
2	1	biimpa 324	. . . . 5 ⊢ ((χ ∧ φ) → ψ)
3	2	19.23aivv 953	. . . 4 ⊢ (∃z∃w(χ ∧ φ) → ψ)
4	3	19.23aivv 953	. . 3 ⊢ (∃x∃y∃z∃w(χ ∧ φ) → ψ)
5	4	a1i 7	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(χ ∧ φ) → ψ))
6	1	biimprcd 138	. . . . . . 7 ⊢ (ψ → (χ → φ))
7	6	ancld 246	. . . . . 6 ⊢ (ψ → (χ → (χ ∧ φ)))
8	7	19.22dvv 949	. . . . 5 ⊢ (ψ → (∃z∃wχ → ∃z∃w(χ ∧ φ)))
9	8	19.22dvv 949	. . . 4 ⊢ (ψ → (∃x∃y∃z∃wχ → ∃x∃y∃z∃w(χ ∧ φ)))
10		elex 1356	. . . . . . . . 9 ⊢ (A ∈ R → ∃x x = A)
11		elex 1356	. . . . . . . . 9 ⊢ (B ∈ S → ∃y y = B)
12	10, 11	anim12i 268	. . . . . . . 8 ⊢ ((A ∈ R ∧ B ∈ S) → (∃x x = A ∧ ∃y y = B))
13		eeanv 980	. . . . . . . 8 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
14	12, 13	sylibr 175	. . . . . . 7 ⊢ ((A ∈ R ∧ B ∈ S) → ∃x∃y(x = A ∧ y = B))
15		elex 1356	. . . . . . . . 9 ⊢ (C ∈ R → ∃z z = C)
16		elex 1356	. . . . . . . . 9 ⊢ (D ∈ S → ∃w w = D)
17	15, 16	anim12i 268	. . . . . . . 8 ⊢ ((C ∈ R ∧ D ∈ S) → (∃z z = C ∧ ∃w w = D))
18		eeanv 980	. . . . . . . 8 ⊢ (∃z∃w(z = C ∧ w = D) ↔ (∃z z = C ∧ ∃w w = D))
19	17, 18	sylibr 175	. . . . . . 7 ⊢ ((C ∈ R ∧ D ∈ S) → ∃z∃w(z = C ∧ w = D))
20	14, 19	anim12i 268	. . . . . 6 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y(x = A ∧ y = B) ∧ ∃z∃w(z = C ∧ w = D)))
21		ee4anv 982	. . . . . 6 ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ↔ (∃x∃y(x = A ∧ y = B) ∧ ∃z∃w(z = C ∧ w = D)))
22	20, 21	sylibr 175	. . . . 5 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → ∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)))
23		cgsex4g.1	. . . . . . . . 9 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → χ)
24	23	19.22i 723	. . . . . . . 8 ⊢ (∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃wχ)
25	24	19.22i 723	. . . . . . 7 ⊢ (∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃z∃wχ)
26	25	19.22i 723	. . . . . 6 ⊢ (∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃y∃z∃wχ)
27	26	19.22i 723	. . . . 5 ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃x∃y∃z∃wχ)
28	22, 27	syl 12	. . . 4 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → ∃x∃y∃z∃wχ)
29	9, 28	syl5 22	. . 3 ⊢ (ψ → (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → ∃x∃y∃z∃w(χ ∧ φ)))
30	29	com12 13	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (ψ → ∃x∃y∃z∃w(χ ∧ φ)))
31	5, 30	impbid 397	1 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(χ ∧ φ) ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  copsex4g 1904  brecop 3242

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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