Theorem cgsex2g 2891

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x,y,ψ   x,A,y   x,B,y

Allowed substitution hints:   φ(x,y)   χ(x,y)   V(x,y)   W(x,y)

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . 4 ⊢ (χ → (φ ↔ ψ))
2	1	biimpa 470	. . 3 ⊢ ((χ ∧ φ) → ψ)
3	2	exlimivv 1635	. 2 ⊢ (∃x∃y(χ ∧ φ) → ψ)
4		elisset 2869	. . . . . 6 ⊢ (A ∈ V → ∃x x = A)
5		elisset 2869	. . . . . 6 ⊢ (B ∈ W → ∃y y = B)
6	4, 5	anim12i 549	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x x = A ∧ ∃y y = B))
7		eeanv 1913	. . . . 5 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
8	6, 7	sylibr 203	. . . 4 ⊢ ((A ∈ V ∧ B ∈ W) → ∃x∃y(x = A ∧ y = B))
9		cgsex2g.1	. . . . 5 ⊢ ((x = A ∧ y = B) → χ)
10	9	2eximi 1577	. . . 4 ⊢ (∃x∃y(x = A ∧ y = B) → ∃x∃yχ)
11	8, 10	syl 15	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → ∃x∃yχ)
12	1	biimprcd 216	. . . . 5 ⊢ (ψ → (χ → φ))
13	12	ancld 536	. . . 4 ⊢ (ψ → (χ → (χ ∧ φ)))
14	13	2eximdv 1624	. . 3 ⊢ (ψ → (∃x∃yχ → ∃x∃y(χ ∧ φ)))
15	11, 14	syl5com 26	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (ψ → ∃x∃y(χ ∧ φ)))
16	3, 15	impbid2 195	1 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x∃y(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 2861

This theorem is referenced by: (None)