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Theorem xorbi12d 1417
Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
xor12d.1  |-  ( ph  ->  ( ps  <->  ch )
)
xor12d.2  |-  ( ph  ->  ( th  <->  ta )
)
Assertion
Ref Expression
xorbi12d  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  ta ) ) )

Proof of Theorem xorbi12d
StepHypRef Expression
1 xor12d.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
2 xor12d.2 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
31, 2bibi12d 322 . . 3  |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  ta ) ) )
43notbid 295 . 2  |-  ( ph  ->  ( -.  ( ps  <->  th )  <->  -.  ( ch  <->  ta ) ) )
5 df-xor 1401 . 2  |-  ( ( ps  \/_  th )  <->  -.  ( ps  <->  th )
)
6 df-xor 1401 . 2  |-  ( ( ch  \/_  ta )  <->  -.  ( ch  <->  ta )
)
74, 5, 63bitr4g 291 1  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  ta ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/_ 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-xor 1401
This theorem is referenced by:  hadbi123d  1493  cadbi123d  1508
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