Theorem xorbi12d 1273

Description: Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)

Hypotheses

Ref	Expression
xorbi12d.1	|- ( ph -> ( ps <-> ch ) )
xorbi12d.2	|- ( ph -> ( th <-> ta ) )

Assertion

Ref	Expression
xorbi12d	|- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )

Proof of Theorem xorbi12d

Step	Hyp	Ref	Expression
1		xorbi12d.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	xorbi1d 1272	. 2 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) )
3		xorbi12d.2	. . 3 |- ( ph -> ( th <-> ta ) )
4	3	xorbi2d 1271	. 2 |- ( ph -> ( ( ch \/_ th ) <-> ( ch \/_ ta ) ) )
5	2, 4	bitrd 177	1 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )

Syntax hints:   -> wi 4   <-> wb 98   \/_ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by:  xorbi12i  1274  anxordi  1291  rpnegap  8615