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Theorem dedth2v 4366
 Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4363 is simpler to use. See also comments in dedth 4362. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth2v.1 (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))
dedth2v.2 (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))
dedth2v.3 𝜃
Assertion
Ref Expression
dedth2v (𝜑𝜓)

Proof of Theorem dedth2v
StepHypRef Expression
1 dedth2v.1 . . 3 (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))
2 dedth2v.2 . . 3 (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))
3 dedth2v.3 . . 3 𝜃
41, 2, 3dedth2h 4363 . 2 ((𝜑𝜑) → 𝜓)
54anidms 562 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601  ifcif 4306 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-if 4307 This theorem is referenced by:  ltweuz  13079  omlsi  28835  pjhfo  29137
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