Theorem vtocleg 2624

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 2568	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1489	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1243  ∃wex 1381   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  vtocle  2627  spsbc  2775  prexgOLD  3946  prexg  3947  funimaexglem  4982  eloprabga  5591  bj-prexg  10031