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Theorem dedth2v 4121
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 4118 is simpler to use. See also comments in dedth 4117. (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 4118 . 2 ((𝜑𝜑) → 𝜓)
54anidms 676 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  ifcif 4064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4065
This theorem is referenced by:  ltweuz  12716  omlsi  28151  pjhfo  28453
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