Theorem cgsex2g 1368

Description: Implicit substitution inference for general classes.

Hypotheses

Ref	Expression
cgsex2g.1	⊢ ((x = A ∧ y = B) → χ)
cgsex2g.2	⊢ (χ → (φ ↔ ψ))

Assertion

Ref	Expression
cgsex2g	⊢ ((A ∈ C ∧ B ∈ D) → (∃x∃y(χ ∧ φ) ↔ ψ))

Distinct variable group(s):   x,y,ψ   x,A,y   x,B,y

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . . 5 ⊢ (χ → (φ ↔ ψ))
2	1	biimpa 324	. . . 4 ⊢ ((χ ∧ φ) → ψ)
3	2	19.23aivv 953	. . 3 ⊢ (∃x∃y(χ ∧ φ) → ψ)
4	3	a1i 7	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x∃y(χ ∧ φ) → ψ))
5	1	biimprcd 138	. . . . . 6 ⊢ (ψ → (χ → φ))
6	5	ancld 246	. . . . 5 ⊢ (ψ → (χ → (χ ∧ φ)))
7	6	19.22dvv 949	. . . 4 ⊢ (ψ → (∃x∃yχ → ∃x∃y(χ ∧ φ)))
8		elex 1356	. . . . . . 7 ⊢ >/FONT>(A ∈ C → ∃x x = A)
9	elex 1356	. . . . . . 7 ⊢ (B ∈ D → ∃y y = B)
10	8, 9	anim12i 268	. . . . . 6 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x x = A ∧ ∃y y = B))
11		eeanv 980	. . . . . 6 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
12	10, 11	sylibr 175	. . . . 5 ⊢ ((A ∈ C ∧ B ∈ D) → ∃x∃y(x = A ∧ y = B))
13		cgsex2g.1	. . . . . . 7 ⊢ ((x = A ∧ y = B) → χ)
14	13	19.22i 723	. . . . . 6 ⊢ (∃y(x = A ∧ y = B) → ∃yχ)
15	14	19.22i 723	. . . . 5 ⊢ (∃x∃y(x = A ∧ y = B) → ∃x∃yχ)
16	12, 15	syl 12	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → ∃x∃yχ)
17	7, 16	syl5 22	. . 3 ⊢ (ψ → ((A ∈ C ∧ B ∈ D) → ∃x∃y(χ ∧ φ)))
18	17	com12 13	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (ψ → ∃x∃y(χ ∧ φ)))
19	4, 18	impbid 397	1 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x∃y(χ ∧ φ) ↔ ψ))

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

This theorem is referenced by:  distrlem5pr 3925

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

metamath.org