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Theorem dedth3v 4283
Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4282. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth3v.1 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓𝜒))
dedth3v.2 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
dedth3v.3 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
dedth3v.4 𝜏
Assertion
Ref Expression
dedth3v (𝜑𝜓)

Proof of Theorem dedth3v
StepHypRef Expression
1 dedth3v.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓𝜒))
2 dedth3v.2 . . . 4 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
3 dedth3v.3 . . . 4 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
4 dedth3v.4 . . . 4 𝜏
51, 2, 3, 4dedth3h 4280 . . 3 ((𝜑𝜑𝜑) → 𝜓)
653anidm12 1529 . 2 ((𝜑𝜑) → 𝜓)
76anidms 556 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  ifcif 4225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-if 4226
This theorem is referenced by:  sseliALT  4925
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