Theorem vtoclf 3475

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2417. (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 3427	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		vtoclf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 221	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1937	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2276	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		vtoclf.4	. 2 ⊢ 𝜑
9	7, 8	mpg 1898	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658  Ⅎwnf 1884   ∈ wcel 2166  Vcvv 3415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417

This theorem is referenced by:  vtoclALT  3477  summolem2a  14824  prodmolem2a  15038  poimirlem24  33978  poimirlem28  33982  monotuz  38350  oddcomabszz  38353  binomcxplemnotnn0  39396  limclner  40679  climinf2mpt  40742  climinfmpt  40743  dvnmptdivc  40949  dvnmul  40954  salpreimagtge  41729  salpreimaltle  41730