Theorem vtocleg 2854

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 2794	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1624	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1375  ∃wex 1518   ∈ wcel 2180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1473  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-v 2781

This theorem is referenced by:  vtocle  2857  spsbc  3020  prexg  4274  funimaexglem  5380  eloprabga  6062  cc3  7422  bj-prexg  16184