Theorem vtoclf 3506

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2393. (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 3456	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		vtoclf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 228	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1838	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2223	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		vtoclf.4	. 2 ⊢ 𝜑
9	7, 8	mpg 1798	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  Vcvv 3441

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-nf 1785  df-clel 2814

This theorem is referenced by:  vtoclALT  3508  summolem2a  15526  prodmolem2a  15743  poimirlem24  35906  poimirlem28  35910  monotuz  41026  oddcomabszz  41029  binomcxplemnotnn0  42295  limclner  43528  climinf2mpt  43591  climinfmpt  43592  dvnmptdivc  43815  dvnmul  43820  salpreimagtge  44600  salpreimaltle  44601