Theorem mdandyvrx3 41151

Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Hypotheses

Ref	Expression
mdandyvrx3.1	|- ( ph \/_ ze )
mdandyvrx3.2	|- ( ps \/_ si )
mdandyvrx3.3	|- ( ch <-> ps )
mdandyvrx3.4	|- ( th <-> ps )
mdandyvrx3.5	|- ( ta <-> ph )
mdandyvrx3.6	|- ( et <-> ph )

Assertion

Ref	Expression
mdandyvrx3	|- ( ( ( ( ch \/_ si ) /\ ( th \/_ si ) ) /\ ( ta \/_ ze ) ) /\ ( et \/_ ze ) )

Proof of Theorem mdandyvrx3

Step	Hyp	Ref	Expression
1		mdandyvrx3.2	. . . . 5 |- ( ps \/_ si )
2		mdandyvrx3.3	. . . . 5 |- ( ch <-> ps )
3	1, 2	axorbciffatcxorb 41072	. . . 4 |- ( ch \/_ si )
4		mdandyvrx3.4	. . . . 5 |- ( th <-> ps )
5	1, 4	axorbciffatcxorb 41072	. . . 4 |- ( th \/_ si )
6	3, 5	pm3.2i 471	. . 3 |- ( ( ch \/_ si ) /\ ( th \/_ si ) )
7		mdandyvrx3.1	. . . 4 |- ( ph \/_ ze )
8		mdandyvrx3.5	. . . 4 |- ( ta <-> ph )
9	7, 8	axorbciffatcxorb 41072	. . 3 |- ( ta \/_ ze )
10	6, 9	pm3.2i 471	. 2 |- ( ( ( ch \/_ si ) /\ ( th \/_ si ) ) /\ ( ta \/_ ze ) )
11		mdandyvrx3.6	. . 3 |- ( et <-> ph )
12	7, 11	axorbciffatcxorb 41072	. 2 |- ( et \/_ ze )
13	10, 12	pm3.2i 471	1 |- ( ( ( ( ch \/_ si ) /\ ( th \/_ si ) ) /\ ( ta \/_ ze ) ) /\ ( et \/_ ze ) )

Syntax hints:   <-> 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:  mdandyvrx12  41160