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Theorem vtoclf 2765
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1737. (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 2720 . . . 4 𝑥 𝑥 = 𝐴
4 vtoclf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 143 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
63, 5eximii 1582 . . 3 𝑥(𝜑𝜓)
71, 619.36i 1652 . 2 (∀𝑥𝜑𝜓)
8 vtoclf.4 . 2 𝜑
97, 8mpg 1431 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wnf 1440  wcel 2128  Vcvv 2712
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  vtocl  2766
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