Theorem vtoclf 3519

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2397. (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 233	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
6	1, 2, 5	vtoclef 3518	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  Vcvv 3438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-clel 2809

This theorem is referenced by:  summolem2a  15636  prodmolem2a  15855  poimirlem24  37784  poimirlem28  37788  monotuz  43125  oddcomabszz  43128  binomcxplemnotnn0  44539  limclner  45837  climinf2mpt  45900  climinfmpt  45901  dvnmptdivc  46124  dvnmul  46129  salpreimagtge  46911  salpreimaltle  46912