Theorem vtoclf 3560

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2413. (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 3510	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		vtoclf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 231	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1837	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2233	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		vtoclf.4	. 2 ⊢ 𝜑
9	7, 8	mpg 1798	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Vcvv 3496

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-cleq 2816  df-clel 2895

This theorem is referenced by:  vtoclALT  3562  summolem2a  15074  prodmolem2a  15290  poimirlem24  34918  poimirlem28  34922  monotuz  39545  oddcomabszz  39548  binomcxplemnotnn0  40695  limclner  41939  climinf2mpt  42002  climinfmpt  42003  dvnmptdivc  42230  dvnmul  42235  salpreimagtge  43009  salpreimaltle  43010