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Theorem elimdhyp 4496
Description: Version of elimhyp 4491 where the hypothesis is deduced from the final antecedent. See divalg 15747 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
Hypotheses
Ref Expression
elimdhyp.1 (𝜑𝜓)
elimdhyp.2 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))
elimdhyp.3 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))
elimdhyp.4 𝜃
Assertion
Ref Expression
elimdhyp 𝜒

Proof of Theorem elimdhyp
StepHypRef Expression
1 elimdhyp.1 . . 3 (𝜑𝜓)
2 iftrue 4434 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2807 . . . 4 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
4 elimdhyp.2 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))
53, 4syl 17 . . 3 (𝜑 → (𝜓𝜒))
61, 5mpbid 235 . 2 (𝜑𝜒)
7 elimdhyp.4 . . 3 𝜃
8 iffalse 4437 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
98eqcomd 2807 . . . 4 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
10 elimdhyp.3 . . . 4 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))
119, 10syl 17 . . 3 𝜑 → (𝜃𝜒))
127, 11mpbii 236 . 2 𝜑𝜒)
136, 12pm2.61i 185 1 𝜒
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  ifcif 4428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-if 4429
This theorem is referenced by:  divalg  15747
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