Theorem vtoclf 3231

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2250. (Contributed by NM, 30-Aug-1993.)

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	. . 3 ⊢ Ⅎ𝑥𝜓
2		vtoclf.2	. . . . 5 ⊢ 𝐴 ∈ V
3	2	isseti 3182	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		vtoclf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 218	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1754	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2086	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		vtoclf.4	. 2 ⊢ 𝜑
9	7, 8	mpg 1715	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  vtoclALT  3233  summolem2a  14293  prodmolem2a  14503  poimirlem24  32603  poimirlem28  32607  monotuz  36524  oddcomabszz  36527  binomcxplemnotnn0  37577  limclner  38718  dvnmptdivc  38828  dvnmul  38833  salpreimagtge  39611  salpreimaltle  39612