Theorem vtocleg 3511

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)

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 2820	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1934	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-clel 2817

This theorem is referenced by:  vtocle  3514  spsbc  3724  prex  5350  avril1  28728  rdgssun  35476  finxpreclem6  35494  ralssiun  35505  frege58c  41418  tz6.12i-afv2  44622