Theorem xorbi12d 1315

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 312	. . 3 |- (ph -> ((ps <-> th) <-> (ch <-> ta)))
4	3	notbid 285	. 2 |- (ph -> (-. (ps <-> th) <-> -. (ch <-> ta)))
5		df-xor 1305	. 2 |- ((ps \/_ th) <-> -. (ps <-> th))
6		df-xor 1305	. 2 |- ((ch \/_ ta) <-> -. (ch <-> ta))
7	4, 5, 6	3bitr4g 279	1 |- (ph -> ((ps \/_ th) <-> (ch \/_ ta)))

Syntax hints:  -. wn 3   -> wi 4   <-> 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:  hadbi123d  1382  cadbi123d  1383