Theorem vtoclef 3528

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 3472	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtoclef.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtoclef.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
5	3, 4	exlimi 2252	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑)
6	2, 5	ax-mp 5	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∃wex 1808  Ⅎwnf 1812   ∈ wcel 2142  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-clel 2837

This theorem is used by:  vtoclf  3529  nn0ind-raph  12702  rdgssun  38052  finxpreclem2  38064