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Theorem vtoclb 2628
 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 228 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
5 vtoclb.4 . 2 (𝜑𝜓)
61, 4, 5vtocl 2625 1 (𝜒𝜃)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259   ∈ wcel 1409  Vcvv 2574 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576 This theorem is referenced by:  alexeq  2693  sbss  3357
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