Theorem vtoclef 3527

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
vtoclef.1	⊢ Ⅎ𝑥𝜑
vtoclef.2	⊢ 𝐴 ∈ V
vtoclef.3	⊢ (𝑥 = 𝐴 → 𝜑)

Assertion

Ref	Expression
vtoclef	⊢ 𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 ⊢ 𝐴 ∈ V
2	1	isseti 3471	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtoclef.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtoclef.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
5	3, 4	exlimi 2255	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
6	2, 5	ax-mp 5	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2145  Vcvv 3453

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  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2837

This theorem is used by:  vtoclf  3528  nn0ind-raph  12722  rdgssun  38117  finxpreclem2  38129