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Theorem dedth3v 4281
Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4280. (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 4278 . . 3 ((𝜑𝜑𝜑) → 𝜓)
653anidm12 1528 . 2 ((𝜑𝜑) → 𝜓)
76anidms 548 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  ifcif 4223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-if 4224
This theorem is referenced by:  sseliALT  4922
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