Theorem abnotbtaxb 41082

Description: Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Hypotheses

Ref	Expression
abnotbtaxb.1	|- ph
abnotbtaxb.2	|- -. ps

Assertion

Ref	Expression
abnotbtaxb	|- ( ph \/_ ps )

Proof of Theorem abnotbtaxb

Step	Hyp	Ref	Expression
1		abnotbtaxb.1	. . 3 |- ph
2		abnotbtaxb.2	. . 3 |- -. ps
3		xor3 372	. . . 4 |- ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) )
4		pm5.1 902	. . . . . 6 |- ( ( ph /\ -. ps ) -> ( ph <-> -. ps ) )
5		ibibr 358	. . . . . 6 |- ( ( ( ph /\ -. ps ) -> ( ph <-> -. ps ) ) <-> ( ( ph /\ -. ps ) -> ( ( ph <-> -. ps ) <-> ( ph /\ -. ps ) ) ) )
6	4, 5	mpbi 220	. . . . 5 |- ( ( ph /\ -. ps ) -> ( ( ph <-> -. ps ) <-> ( ph /\ -. ps ) ) )
7	1, 2, 6	mp2an 708	. . . 4 |- ( ( ph <-> -. ps ) <-> ( ph /\ -. ps ) )
8	3, 7	bitri 264	. . 3 |- ( -. ( ph <-> ps ) <-> ( ph /\ -. ps ) )
9	1, 2, 8	mpbir2an 955	. 2 |- -. ( ph <-> ps )
10		df-xor 1465	. 2 |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
11	9, 10	mpbir 221	1 |- ( ph \/_ ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/_ 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-an 386  df-xor 1465

This theorem is referenced by:  aistbisfiaxb  41086  aifftbifffaibifff  41089