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Theorem vtoclb 2717
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
Hypotheses
Ref Expression
vtoclb.1 𝐴 ∈ V
vtoclb.2 (𝑥 = 𝐴 → (𝜑𝜒))
vtoclb.3 (𝑥 = 𝐴 → (𝜓𝜃))
vtoclb.4 (𝜑𝜓)
Assertion
Ref Expression
vtoclb (𝜒𝜃)
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclb
StepHypRef Expression
1 vtoclb.1 . 2 𝐴 ∈ V
2 vtoclb.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜒))
3 vtoclb.3 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
42, 3bibi12d 234 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
5 vtoclb.4 . 2 (𝜑𝜓)
61, 4, 5vtocl 2714 1 (𝜒𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  Vcvv 2660
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  alexeq  2785  sbss  3441
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