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Theorem xorbi2d 1316
Description: Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
Hypothesis
Ref Expression
xorbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
xorbi2d  |-  ( ph  ->  ( ( th  \/_  ps )  <->  ( th  \/_  ch ) ) )

Proof of Theorem xorbi2d
StepHypRef Expression
1 xorbid.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
21orbi2d 739 . . 3  |-  ( ph  ->  ( ( th  \/  ps )  <->  ( th  \/  ch ) ) )
31anbi2d 452 . . . 4  |-  ( ph  ->  ( ( th  /\  ps )  <->  ( th  /\  ch ) ) )
43notbid 627 . . 3  |-  ( ph  ->  ( -.  ( th 
/\  ps )  <->  -.  ( th  /\  ch ) ) )
52, 4anbi12d 457 . 2  |-  ( ph  ->  ( ( ( th  \/  ps )  /\  -.  ( th  /\  ps ) )  <->  ( ( th  \/  ch )  /\  -.  ( th  /\  ch ) ) ) )
6 df-xor 1312 . 2  |-  ( ( th  \/_  ps )  <->  ( ( th  \/  ps )  /\  -.  ( th 
/\  ps ) ) )
7 df-xor 1312 . 2  |-  ( ( th  \/_  ch )  <->  ( ( th  \/  ch )  /\  -.  ( th 
/\  ch ) ) )
85, 6, 73bitr4g 221 1  |-  ( ph  ->  ( ( th  \/_  ps )  <->  ( th  \/_  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    \/_ wxo 1311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-xor 1312
This theorem is referenced by:  xorbi12d  1318
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