Theorem vtoclf 3530

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2427. (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 236	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
6	1, 2, 5	vtoclef 3529	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  Vcvv 3455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-clel 2838

This theorem is used by:  summolem2a  15771  prodmolem2a  15993  poimirlem24  38323  poimirlem28  38327  monotuz  43696  oddcomabszz  43699  binomcxplemnotnn0  45094  limclner  46393  climinf2mpt  46456  climinfmpt  46457  dvnmptdivc  46680  dvnmul  46685  salpreimagtge  47467  salpreimaltle  47468