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Theorem elimhyp 4596
Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4589. (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 4537 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2741 . . . 4 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 elimhyp.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))
42, 3syl 17 . . 3 (𝜑 → (𝜑𝜓))
54ibi 267 . 2 (𝜑𝜓)
6 elimhyp.3 . . 3 𝜒
7 iffalse 4540 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2741 . . . 4 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 elimhyp.2 . . . 4 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))
108, 9syl 17 . . 3 𝜑 → (𝜒𝜓))
116, 10mpbii 233 . 2 𝜑𝜓)
125, 11pm2.61i 182 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  ifcif 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532
This theorem is referenced by:  elimel  4600  elimf  6736  oeoa  8634  oeoe  8636  limensuc  9193  axcc4dom  10479  elimne0  11249  elimgt0  12103  elimge0  12104  2ndcdisj  23480  siilem2  30881  normlem7tALT  31148  hhsssh  31298  shintcl  31359  chintcl  31361  spanun  31574  elunop2  32042  lnophm  32048  nmbdfnlb  32079  hmopidmch  32182  hmopidmpj  32183  chirred  32424  limsucncmp  36429  elimhyps  38943  elimhyps2  38946
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