MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3anbi2i Structured version   Visualization version   GIF version

Theorem 3anbi2i 1161
Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
Hypothesis
Ref Expression
3anbi1i.1 (𝜑𝜓)
Assertion
Ref Expression
3anbi2i ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))

Proof of Theorem 3anbi2i
StepHypRef Expression
1 biid 251 . 2 (𝜒𝜒)
2 3anbi1i.1 . 2 (𝜑𝜓)
3 biid 251 . 2 (𝜃𝜃)
41, 2, 33anbi123i 1158 1 ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1073
This theorem is referenced by:  f13dfv  6672  axgroth4  9855  fi1uzind  13480  bnj543  31298  bnj916  31338  topdifinffinlem  33528
  Copyright terms: Public domain W3C validator