Theorem xorbi2d 1370

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

Step	Hyp	Ref	Expression
1		xorbid.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2	1	orbi2d 780	. . 3 |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) )
3	1	anbi2d 460	. . . 4 |- ( ph -> ( ( th /\ ps ) <-> ( th /\ ch ) ) )
4	3	notbid 657	. . 3 |- ( ph -> ( -. ( th /\ ps ) <-> -. ( th /\ ch ) ) )
5	2, 4	anbi12d 465	. 2 |- ( ph -> ( ( ( th \/ ps ) /\ -. ( th /\ ps ) ) <-> ( ( th \/ ch ) /\ -. ( th /\ ch ) ) ) )
6		df-xor 1366	. 2 |- ( ( th \/_ ps ) <-> ( ( th \/ ps ) /\ -. ( th /\ ps ) ) )
7		df-xor 1366	. 2 |- ( ( th \/_ ch ) <-> ( ( th \/ ch ) /\ -. ( th /\ ch ) ) )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( ( th \/_ ps ) <-> ( th \/_ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   \/_ wxo 1365

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by:  xorbi12d  1372