MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elimdhyp Structured version   Visualization version   GIF version

Theorem elimdhyp 4531
Description: Version of elimhyp 4526 where the hypothesis is deduced from the final antecedent. See divalg 15742 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 4469 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2824 . . . 4 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
4 elimdhyp.2 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))
53, 4syl 17 . . 3 (𝜑 → (𝜓𝜒))
61, 5mpbid 233 . 2 (𝜑𝜒)
7 elimdhyp.4 . . 3 𝜃
8 iffalse 4472 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
98eqcomd 2824 . . . 4 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
10 elimdhyp.3 . . . 4 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))
119, 10syl 17 . . 3 𝜑 → (𝜃𝜒))
127, 11mpbii 234 . 2 𝜑𝜒)
136, 12pm2.61i 183 1 𝜒
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207   = wceq 1528  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-if 4464
This theorem is referenced by:  divalg  15742
  Copyright terms: Public domain W3C validator