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Theorem elimdhyp 4101
 Description: Version of elimhyp 4096 where the hypothesis is deduced from the final antecedent. See divalg 14964 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 4042 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2616 . . . 4 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
4 elimdhyp.2 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))
53, 4syl 17 . . 3 (𝜑 → (𝜓𝜒))
61, 5mpbid 221 . 2 (𝜑𝜒)
7 elimdhyp.4 . . 3 𝜃
8 iffalse 4045 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
98eqcomd 2616 . . . 4 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
10 elimdhyp.3 . . . 4 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))
119, 10syl 17 . . 3 𝜑 → (𝜃𝜒))
127, 11mpbii 222 . 2 𝜑𝜒)
136, 12pm2.61i 175 1 𝜒
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  divalg  14964
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