Theorem xorbi12d 1478

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

Step	Hyp	Ref	Expression
1		xor12d.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2		xor12d.2	. . . 4 |- ( ph -> ( th <-> ta ) )
3	1, 2	bibi12d 335	. . 3 |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) )
4	3	notbid 308	. 2 |- ( ph -> ( -. ( ps <-> th ) <-> -. ( ch <-> ta ) ) )
5		df-xor 1465	. 2 |- ( ( ps \/_ th ) <-> -. ( ps <-> th ) )
6		df-xor 1465	. 2 |- ( ( ch \/_ ta ) <-> -. ( ch <-> ta ) )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/_ wxo 1464

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 197  df-xor 1465

This theorem is referenced by:  hadbi123d  1534  cadbi123d  1549