Theorem xorbi12i 1314

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 306	. . 3 |- ((ph <-> ch) <-> (ps <-> th))
4	3	notbii 287	. 2 |- (-. (ph <-> ch) <-> -. (ps <-> th))
5		df-xor 1305	. 2 |- ((ph \/_ ch) <-> -. (ph <-> ch))
6		df-xor 1305	. 2 |- ((ps \/_ th) <-> -. (ps <-> th))
7	4, 5, 6	3bitr4i 268	1 |- ((ph \/_ ch) <-> (ps \/_ th))

Syntax hints:  -. wn 3   <-> wb 176   \/_ wxo 1304

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 177  df-xor 1305

This theorem is referenced by:  hadcoma  1388  hadcomb  1389  hadnot  1393