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Theorem elimhyp 4306
Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4299. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
elimhyp.1 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))
elimhyp.2 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))
elimhyp.3 𝜒
Assertion
Ref Expression
elimhyp 𝜓

Proof of Theorem elimhyp
StepHypRef Expression
1 iftrue 4249 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2771 . . . 4 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 elimhyp.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))
42, 3syl 17 . . 3 (𝜑 → (𝜑𝜓))
54ibi 258 . 2 (𝜑𝜓)
6 elimhyp.3 . . 3 𝜒
7 iffalse 4252 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2771 . . . 4 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 elimhyp.2 . . . 4 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))
108, 9syl 17 . . 3 𝜑 → (𝜒𝜓))
116, 10mpbii 224 . 2 𝜑𝜓)
125, 11pm2.61i 176 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1652  ifcif 4243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-if 4244
This theorem is referenced by:  elimel  4310  elimf  6222  oeoa  7882  oeoe  7884  limensuc  8344  axcc4dom  9516  elimne0  10283  elimgt0  11113  elimge0  11114  2ndcdisj  21539  siilem2  28163  normlem7tALT  28432  hhsssh  28582  shintcl  28645  chintcl  28647  spanun  28860  elunop2  29328  lnophm  29334  nmbdfnlb  29365  hmopidmch  29468  hmopidmpj  29469  chirred  29710  limsucncmp  32884  elimhyps  34917  elimhyps2  34920
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