Theorem vtocleg 3581

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 3505	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1931	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1780   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  vtocle  3584  spsbc  3785  prex  5333  avril1  28242  rdgssun  34662  finxpreclem6  34680  ralssiun  34691  frege58c  40287  tz6.12i-afv2  43462