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Theorem keephyp 4102
 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 4074 . 2 ((𝜓𝜒) → 𝜃)
61, 2, 5mp2an 704 1 𝜃
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  keepel  4105  boxcutc  7837  fin23lem13  9037  abvtrivd  18663  znf1o  19719  zntoslem  19724  dscmet  22187  sqff1o  24708  lgsne0  24860  dchrisum0flblem1  24997  dchrisum0flblem2  24998
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