Theorem vtocleg 2926

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)

Hypothesis

Ref	Expression
vtocleg.1	⊢ (x = A → φ)

Assertion

Ref	Expression
vtocleg	⊢ (A ∈ V → φ)

Distinct variable groups:   x,A   φ,x

Allowed substitution hint:   V(x)

Proof of Theorem vtocleg

Step	Hyp	Ref	Expression
1		elisset 2870	. 2 ⊢ (A ∈ V → ∃x x = A)
2		vtocleg.1	. . 3 ⊢ (x = A → φ)
3	2	exlimiv 1634	. 2 ⊢ (∃x x = A → φ)
4	1, 3	syl 15	1 ⊢ (A ∈ V → φ)

Syntax hints:   → wi 4  ∃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:  vtocle  2929  spsbc  3059  eloprabga  5579