MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm2.61iine Structured version   Unicode version

Theorem pm2.61iine 2707
Description: Equality version of pm2.61ii 165. (Contributed by Scott Fenton, 13-Jun-2013.)
Hypotheses
Ref Expression
pm2.61iine.1  |-  ( ( A  =/=  C  /\  B  =/=  D )  ->  ph )
pm2.61iine.2  |-  ( A  =  C  ->  ph )
pm2.61iine.3  |-  ( B  =  D  ->  ph )
Assertion
Ref Expression
pm2.61iine  |-  ph

Proof of Theorem pm2.61iine
StepHypRef Expression
1 df-ne 2622 . . . 4  |-  ( A  =/=  C  <->  -.  A  =  C )
2 df-ne 2622 . . . 4  |-  ( B  =/=  D  <->  -.  B  =  D )
3 pm2.61iine.1 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  D )  ->  ph )
41, 2, 3syl2anbr 480 . . 3  |-  ( ( -.  A  =  C  /\  -.  B  =  D )  ->  ph )
54ex 434 . 2  |-  ( -.  A  =  C  -> 
( -.  B  =  D  ->  ph ) )
6 pm2.61iine.2 . 2  |-  ( A  =  C  ->  ph )
7 pm2.61iine.3 . 2  |-  ( B  =  D  ->  ph )
85, 6, 7pm2.61ii 165 1  |-  ph
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    =/= wne 2620
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-an 371  df-ne 2622
This theorem is referenced by:  dedekind  9548
  Copyright terms: Public domain W3C validator