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Theorem vtoclb 3294
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 334 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
5 vtoclb.4 . 2 (𝜑𝜓)
61, 4, 5vtocl 3290 1 (𝜒𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  sbss  4117  bnj609  31113
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