Theorem vtoclf 2779

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1745. (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 2734	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		vtoclf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 143	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1590	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 1660	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		vtoclf.4	. 2 ⊢ 𝜑
9	7, 8	mpg 1439	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  vtocl  2780