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Theorem xorbi1d 1288
Description: Deduction joining an equivalence and a right 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
xorbi1d  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  th ) ) )

Proof of Theorem xorbi1d
StepHypRef Expression
1 xorbid.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
21orbi1d 715 . . 3  |-  ( ph  ->  ( ( ps  \/  th )  <->  ( ch  \/  th ) ) )
31anbi1d 446 . . . 4  |-  ( ph  ->  ( ( ps  /\  th )  <->  ( ch  /\  th ) ) )
43notbid 602 . . 3  |-  ( ph  ->  ( -.  ( ps 
/\  th )  <->  -.  ( ch  /\  th ) ) )
52, 4anbi12d 450 . 2  |-  ( ph  ->  ( ( ( ps  \/  th )  /\  -.  ( ps  /\  th ) )  <->  ( ( ch  \/  th )  /\  -.  ( ch  /\  th ) ) ) )
6 df-xor 1283 . 2  |-  ( ( ps  \/_  th )  <->  ( ( ps  \/  th )  /\  -.  ( ps 
/\  th ) ) )
7 df-xor 1283 . 2  |-  ( ( ch  \/_  th )  <->  ( ( ch  \/  th )  /\  -.  ( ch 
/\  th ) ) )
85, 6, 73bitr4g 216 1  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    \/_ wxo 1282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-xor 1283
This theorem is referenced by:  xorbi12d  1289
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