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Theorem keephyp 4595
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 4563 . 2 ((𝜓𝜒) → 𝜃)
61, 2, 5mp2an 691 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  ifcif 4524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-if 4525
This theorem is referenced by:  boxcutc  8953  fin23lem13  10349  abvtrivd  20713  znf1o  21478  zntoslem  21483  dscmet  24474  sqff1o  27107  lgsne0  27261  dchrisum0flblem1  27434  dchrisum0flblem2  27435
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