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Theorem keephyp 4312
Description: Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
keephyp.1 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
keephyp.2 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
keephyp.3 𝜓
keephyp.4 𝜒
Assertion
Ref Expression
keephyp 𝜃

Proof of Theorem keephyp
StepHypRef Expression
1 keephyp.3 . 2 𝜓
2 keephyp.4 . 2 𝜒
3 keephyp.1 . . 3 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
4 keephyp.2 . . 3 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
53, 4ifboth 4281 . 2 ((𝜓𝜒) → 𝜃)
61, 2, 5mp2an 683 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  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 2069  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 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-if 4244
This theorem is referenced by:  boxcutc  8156  fin23lem13  9407  abvtrivd  19109  znf1o  20172  zntoslem  20177  dscmet  22656  sqff1o  25199  lgsne0  25351  dchrisum0flblem1  25488  dchrisum0flblem2  25489
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