Theorem vtoclf 3548

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2395. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)

Hypotheses

Ref	Expression
vtoclf.1	⊢ Ⅎ𝑥𝜓
vtoclf.2	⊢ 𝐴 ∈ V
vtoclf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclf.4	⊢ 𝜑

Assertion

Ref	Expression
vtoclf	⊢ 𝜓

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. 2 ⊢ Ⅎ𝑥𝜓
2		vtoclf.2	. 2 ⊢ 𝐴 ∈ V
3		vtoclf.4	. . 3 ⊢ 𝜑
4		vtoclf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpbii 232	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
6	1, 2, 5	vtoclef 3547	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-clel 2811

This theorem is referenced by:  vtoclALT  3551  summolem2a  15661  prodmolem2a  15878  poimirlem24  36560  poimirlem28  36564  monotuz  41728  oddcomabszz  41731  binomcxplemnotnn0  43163  limclner  44415  climinf2mpt  44478  climinfmpt  44479  dvnmptdivc  44702  dvnmul  44707  salpreimagtge  45489  salpreimaltle  45490