Theorem xorbi12d 1382

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 1381	. 2 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) )
3		xorbi12d.2	. . 3 |- ( ph -> ( th <-> ta ) )
4	3	xorbi2d 1380	. 2 |- ( ph -> ( ( ch \/_ th ) <-> ( ch \/_ ta ) ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/_ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-xor 1376

This theorem is referenced by:  xorbi12i  1383  anxordi  1400  rpnegap  9684