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Theorem abnotbtaxb 37902
Description: Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
Hypotheses
Ref Expression
abnotbtaxb.1  |-  ph
abnotbtaxb.2  |-  -.  ps
Assertion
Ref Expression
abnotbtaxb  |-  ( ph  \/_ 
ps )

Proof of Theorem abnotbtaxb
StepHypRef Expression
1 abnotbtaxb.1 . . 3  |-  ph
2 abnotbtaxb.2 . . 3  |-  -.  ps
3 xor3 358 . . . 4  |-  ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )
4 pm5.1 865 . . . . . 6  |-  ( (
ph  /\  -.  ps )  ->  ( ph  <->  -.  ps )
)
5 ibibr 344 . . . . . 6  |-  ( ( ( ph  /\  -.  ps )  ->  ( ph  <->  -. 
ps ) )  <->  ( ( ph  /\  -.  ps )  ->  ( ( ph  <->  -.  ps )  <->  (
ph  /\  -.  ps )
) ) )
64, 5mpbi 211 . . . . 5  |-  ( (
ph  /\  -.  ps )  ->  ( ( ph  <->  -.  ps )  <->  (
ph  /\  -.  ps )
) )
71, 2, 6mp2an 676 . . . 4  |-  ( (
ph 
<->  -.  ps )  <->  ( ph  /\ 
-.  ps ) )
83, 7bitri 252 . . 3  |-  ( -.  ( ph  <->  ps )  <->  (
ph  /\  -.  ps )
)
91, 2, 8mpbir2an 928 . 2  |-  -.  ( ph 
<->  ps )
10 df-xor 1401 . 2  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
119, 10mpbir 212 1  |-  ( ph  \/_ 
ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    \/_ wxo 1400
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 188  df-an 372  df-xor 1401
This theorem is referenced by:  aistbisfiaxb  37906  aifftbifffaibifff  37909
  Copyright terms: Public domain W3C validator