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Theorem keephyp 4497
 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 4466 . 2 ((𝜓𝜒) → 𝜃)
61, 2, 5mp2an 691 1 𝜃
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ifcif 4428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-if 4429 This theorem is referenced by:  boxcutc  8492  fin23lem13  9747  abvtrivd  19608  znf1o  20247  zntoslem  20252  dscmet  23183  sqff1o  25771  lgsne0  25923  dchrisum0flblem1  26096  dchrisum0flblem2  26097
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