Theorem vtocleg 3522

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 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1930	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-clel 2804

This theorem is referenced by:  vtoclg  3523  vtocleOLD  3525  spsbc  3769  snexg  5393  prex  5395  avril1  30399  bj-snexg  37029  rdgssun  37373  finxpreclem6  37391  ralssiun  37402  frege58c  43917  tz6.12i-afv2  47248