Theorem xorbi12i 1477

Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Hypotheses

Ref	Expression
xorbi12.1	|- ( ph <-> ps )
xorbi12.2	|- ( ch <-> th )

Assertion

Ref	Expression
xorbi12i	|- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )

Proof of Theorem xorbi12i

Step	Hyp	Ref	Expression
1		xorbi12.1	. . . 4 |- ( ph <-> ps )
2		xorbi12.2	. . . 4 |- ( ch <-> th )
3	1, 2	bibi12i 329	. . 3 |- ( ( ph <-> ch ) <-> ( ps <-> th ) )
4	3	notbii 310	. 2 |- ( -. ( ph <-> ch ) <-> -. ( ps <-> th ) )
5		df-xor 1465	. 2 |- ( ( ph \/_ ch ) <-> -. ( ph <-> ch ) )
6		df-xor 1465	. 2 |- ( ( ps \/_ th ) <-> -. ( ps <-> th ) )
7	4, 5, 6	3bitr4i 292	1 |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )

Syntax hints:   -. wn 3   <-> 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:  hadcoma  1538  hadcomb  1539