Theorem cgsexg 2891

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Hypotheses

Ref	Expression
cgsexg.1	⊢ (x = A → χ)
cgsexg.2	⊢ (χ → (φ ↔ ψ))

Assertion

Ref	Expression
cgsexg	⊢ (A ∈ V → (∃x(χ ∧ φ) ↔ ψ))

Distinct variable groups:   x,A   ψ,x

Allowed substitution hints:   φ(x)   χ(x)   V(x)

Proof of Theorem cgsexg

Step	Hyp	Ref	Expression
1		cgsexg.2	. . . 4 ⊢ (χ → (φ ↔ ψ))
2	1	biimpa 470	. . 3 ⊢ ((χ ∧ φ) → ψ)
3	2	exlimiv 1634	. 2 ⊢ (∃x(χ ∧ φ) → ψ)
4		elisset 2870	. . . 4 ⊢ (A ∈ V → ∃x x = A)
5		cgsexg.1	. . . . 5 ⊢ (x = A → χ)
6	5	eximi 1576	. . . 4 ⊢ (∃x x = A → ∃xχ)
7	4, 6	syl 15	. . 3 ⊢ (A ∈ V → ∃xχ)
8	1	biimprcd 216	. . . . 5 ⊢ (ψ → (χ → φ))
9	8	ancld 536	. . . 4 ⊢ (ψ → (χ → (χ ∧ φ)))
10	9	eximdv 1622	. . 3 ⊢ (ψ → (∃xχ → ∃x(χ ∧ φ)))
11	7, 10	syl5com 26	. 2 ⊢ (A ∈ V → (ψ → ∃x(χ ∧ φ)))
12	3, 11	impbid2 195	1 ⊢ (A ∈ V → (∃x(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-11 1746  ax-ext 2334

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

This theorem is referenced by: (None)