Theorem xorbi1d 1359

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

Step	Hyp	Ref	Expression
1		xorbid.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2	1	orbi1d 780	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) )
3	1	anbi1d 460	. . . 4 |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) )
4	3	notbid 656	. . 3 |- ( ph -> ( -. ( ps /\ th ) <-> -. ( ch /\ th ) ) )
5	2, 4	anbi12d 464	. 2 |- ( ph -> ( ( ( ps \/ th ) /\ -. ( ps /\ th ) ) <-> ( ( ch \/ th ) /\ -. ( ch /\ th ) ) ) )
6		df-xor 1354	. 2 |- ( ( ps \/_ th ) <-> ( ( ps \/ th ) /\ -. ( ps /\ th ) ) )
7		df-xor 1354	. 2 |- ( ( ch \/_ th ) <-> ( ( ch \/ th ) /\ -. ( ch /\ th ) ) )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   \/_ wxo 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-xor 1354

This theorem is referenced by:  xorbi12d  1360