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Theorem vtoclf 2711
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1713. (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 2666 . . . 4 𝑥 𝑥 = 𝐴
4 vtoclf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 143 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
63, 5eximii 1564 . . 3 𝑥(𝜑𝜓)
71, 619.36i 1633 . 2 (∀𝑥𝜑𝜓)
8 vtoclf.4 . 2 𝜑
97, 8mpg 1410 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnf 1419  wcel 1463  Vcvv 2658
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  vtocl  2712
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