Theorem vtocleg 3516

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 2842	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtocleg.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
3	2	exlimiv 1963	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
4	1, 3	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → 𝜑)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-clel 2835

This theorem is used by:  vtoclg  3517  spsbc  3752  snexgALT  5406  prexOLD  5408  avril1  30943  bj-snexg  37778  rdgssun  38132  finxpreclem6  38150  ralssiun  38161  frege58c  44761  tz6.12i-afv2  48131