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Theorem vtoclf 3398
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2407. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtoclf.1 𝑥𝜓
vtoclf.2 𝐴 ∈ V
vtoclf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclf.4 𝜑
Assertion
Ref Expression
vtoclf 𝜓
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclf
StepHypRef Expression
1 vtoclf.1 . . 3 𝑥𝜓
2 vtoclf.2 . . . . 5 𝐴 ∈ V
32isseti 3349 . . . 4 𝑥 𝑥 = 𝐴
4 vtoclf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 219 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
63, 5eximii 1913 . . 3 𝑥(𝜑𝜓)
71, 619.36i 2246 . 2 (∀𝑥𝜑𝜓)
8 vtoclf.4 . 2 𝜑
97, 8mpg 1873 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wnf 1857  wcel 2139  Vcvv 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by:  vtoclALT  3400  summolem2a  14665  prodmolem2a  14883  poimirlem24  33764  poimirlem28  33768  monotuz  38026  oddcomabszz  38029  binomcxplemnotnn0  39075  limclner  40404  climinf2mpt  40467  climinfmpt  40468  dvnmptdivc  40674  dvnmul  40679  salpreimagtge  41458  salpreimaltle  41459
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