MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  keephyp Structured version   Visualization version   GIF version

Theorem keephyp 4419
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 4388 . 2 ((𝜓𝜒) → 𝜃)
61, 2, 5mp2an 679 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  ifcif 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-if 4351
This theorem is referenced by:  boxcutc  8302  fin23lem13  9552  abvtrivd  19333  znf1o  20400  zntoslem  20405  dscmet  22885  sqff1o  25461  lgsne0  25613  dchrisum0flblem1  25786  dchrisum0flblem2  25787
  Copyright terms: Public domain W3C validator