Theorem vtocleg 2669

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 2613	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1529	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1284  ∃wex 1421   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  vtocle  2672  spsbc  2826  prexg  3966  funimaexglem  5002  eloprabga  5611  bj-prexg  10702