MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  jaao Structured version   Unicode version

Theorem jaao 507
Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
Hypotheses
Ref Expression
jaao.1  |-  ( ph  ->  ( ps  ->  ch ) )
jaao.2  |-  ( th 
->  ( ta  ->  ch ) )
Assertion
Ref Expression
jaao  |-  ( (
ph  /\  th )  ->  ( ( ps  \/  ta )  ->  ch )
)

Proof of Theorem jaao
StepHypRef Expression
1 jaao.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21adantr 463 . 2  |-  ( (
ph  /\  th )  ->  ( ps  ->  ch ) )
3 jaao.2 . . 3  |-  ( th 
->  ( ta  ->  ch ) )
43adantl 464 . 2  |-  ( (
ph  /\  th )  ->  ( ta  ->  ch ) )
52, 4jaod 378 1  |-  ( (
ph  /\  th )  ->  ( ( ps  \/  ta )  ->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367
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 185  df-or 368  df-an 369
This theorem is referenced by:  pm3.44  509  pm3.48  834  prlem1  963  ordtri1  5442  ordun  5510  suc11  5512  funun  5610  poxp  6895  suc11reg  8068  rankunb  8299  gruun  9213  ofpreima2  27937  wl-orel12  31321
  Copyright terms: Public domain W3C validator