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Theorem abnotbtaxb 27987
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 . . . 4  |-  ph
2 abnotbtaxb.2 . . . 4  |-  -.  ps
31, 2pm3.2i 441 . . 3  |-  ( ph  /\ 
-.  ps )
4 xor3 346 . . . . . . 7  |-  ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )
5 pm5.1 830 . . . . . . . . 9  |-  ( (
ph  /\  -.  ps )  ->  ( ph  <->  -.  ps )
)
6 ibibr 332 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  ps )  ->  ( ph  <->  -. 
ps ) )  <->  ( ( ph  /\  -.  ps )  ->  ( ( ph  <->  -.  ps )  <->  (
ph  /\  -.  ps )
) ) )
76biimpi 186 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  ps )  ->  ( ph  <->  -. 
ps ) )  -> 
( ( ph  /\  -.  ps )  ->  (
( ph  <->  -.  ps )  <->  (
ph  /\  -.  ps )
) ) )
85, 7ax-mp 8 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  ( ( ph  <->  -.  ps )  <->  (
ph  /\  -.  ps )
) )
93, 8ax-mp 8 . . . . . . 7  |-  ( (
ph 
<->  -.  ps )  <->  ( ph  /\ 
-.  ps ) )
104, 9pm3.2i 441 . . . . . 6  |-  ( ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )  /\  ( ( ph  <->  -. 
ps )  <->  ( ph  /\ 
-.  ps ) ) )
11 bitr 689 . . . . . 6  |-  ( ( ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
)  /\  ( ( ph 
<->  -.  ps )  <->  ( ph  /\ 
-.  ps ) ) )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  /\ 
-.  ps ) ) )
1210, 11ax-mp 8 . . . . 5  |-  ( -.  ( ph  <->  ps )  <->  (
ph  /\  -.  ps )
)
13 bicom 191 . . . . . 6  |-  ( ( -.  ( ph  <->  ps )  <->  (
ph  /\  -.  ps )
)  <->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph 
<->  ps ) ) )
1413biimpi 186 . . . . 5  |-  ( ( -.  ( ph  <->  ps )  <->  (
ph  /\  -.  ps )
)  ->  ( ( ph  /\  -.  ps )  <->  -.  ( ph  <->  ps )
) )
1512, 14ax-mp 8 . . . 4  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  <->  ps )
)
1615biimpi 186 . . 3  |-  ( (
ph  /\  -.  ps )  ->  -.  ( ph  <->  ps )
)
173, 16ax-mp 8 . 2  |-  -.  ( ph 
<->  ps )
18 df-xor 1296 . . . 4  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
19 bicom 191 . . . . 5  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  <->  ( -.  ( ph 
<->  ps )  <->  ( ph  \/_ 
ps ) ) )
2019biimpi 186 . . . 4  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  ->  ( -.  ( ph  <->  ps )  <->  ( ph  \/_ 
ps ) ) )
2118, 20ax-mp 8 . . 3  |-  ( -.  ( ph  <->  ps )  <->  (
ph  \/_  ps )
)
2221biimpi 186 . 2  |-  ( -.  ( ph  <->  ps )  ->  ( ph  \/_  ps ) )
2317, 22ax-mp 8 1  |-  ( ph  \/_ 
ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/_ wxo 1295
This theorem is referenced by:  aistbisfiaxb  27991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-xor 1296
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