Theorem cgsex4g 2893

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 470	. . . 4 ⊢ ((χ ∧ φ) → ψ)
3	2	exlimivv 1635	. . 3 ⊢ (∃z∃w(χ ∧ φ) → ψ)
4	3	exlimivv 1635	. 2 ⊢ (∃x∃y∃z∃w(χ ∧ φ) → ψ)
5		elisset 2870	. . . . . . . 8 ⊢ (A ∈ R → ∃x x = A)
6		elisset 2870	. . . . . . . 8 ⊢ (B ∈ S → ∃y y = B)
7	5, 6	anim12i 549	. . . . . . 7 ⊢ ((A ∈ R ∧ B ∈ S) → (∃x x = A ∧ ∃y y = B))
8		eeanv 1913	. . . . . . 7 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
9	7, 8	sylibr 203	. . . . . 6 ⊢ ((A ∈ R ∧ B ∈ S) → ∃x∃y(x = A ∧ y = B))
10		elisset 2870	. . . . . . . 8 ⊢ (C ∈ R → ∃z z = C)
11		elisset 2870	. . . . . . . 8 ⊢ (D ∈ S → ∃w w = D)
12	10, 11	anim12i 549	. . . . . . 7 ⊢ ((C ∈ R ∧ D ∈ S) → (∃z z = C ∧ ∃w w = D))
13		eeanv 1913	. . . . . . 7 ⊢ (∃z∃w(z = C ∧ w = D) ↔ (∃z z = C ∧ ∃w w = D))
14	12, 13	sylibr 203	. . . . . 6 ⊢ ((C ∈ R ∧ D ∈ S) → ∃z∃w(z = C ∧ w = D))
15	9, 14	anim12i 549	. . . . 5 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y(x = A ∧ y = B) ∧ ∃z∃w(z = C ∧ w = D)))
16		ee4anv 1915	. . . . 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 203	. . . 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 1577	. . . . 5 ⊢ (∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃z∃wχ)
20	19	2eximi 1577	. . . 4 ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ∃x∃y∃z∃wχ)
21	17, 20	syl 15	. . 3 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → ∃x∃y∃z∃wχ)
22	1	biimprcd 216	. . . . . 6 ⊢ (ψ → (χ → φ))
23	22	ancld 536	. . . . 5 ⊢ (ψ → (χ → (χ ∧ φ)))
24	23	2eximdv 1624	. . . 4 ⊢ (ψ → (∃z∃wχ → ∃z∃w(χ ∧ φ)))
25	24	2eximdv 1624	. . 3 ⊢ (ψ → (∃x∃y∃z∃wχ → ∃x∃y∃z∃w(χ ∧ φ)))
26	21, 25	syl5com 26	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (ψ → ∃x∃y∃z∃w(χ ∧ φ)))
27	4, 26	impbid2 195	1 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(χ ∧ φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  copsex4g  4611