Theorem mdandyvrx4 41152

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
mdandyvrx4.1	|- ( ph \/_ ze )
mdandyvrx4.2	|- ( ps \/_ si )
mdandyvrx4.3	|- ( ch <-> ph )
mdandyvrx4.4	|- ( th <-> ph )
mdandyvrx4.5	|- ( ta <-> ps )
mdandyvrx4.6	|- ( et <-> ph )

Assertion

Ref	Expression
mdandyvrx4	|- ( ( ( ( ch \/_ ze ) /\ ( th \/_ ze ) ) /\ ( ta \/_ si ) ) /\ ( et \/_ ze ) )

Proof of Theorem mdandyvrx4

Step	Hyp	Ref	Expression
1		mdandyvrx4.1	. . . . 5 |- ( ph \/_ ze )
2		mdandyvrx4.3	. . . . 5 |- ( ch <-> ph )
3	1, 2	axorbciffatcxorb 41072	. . . 4 |- ( ch \/_ ze )
4		mdandyvrx4.4	. . . . 5 |- ( th <-> ph )
5	1, 4	axorbciffatcxorb 41072	. . . 4 |- ( th \/_ ze )
6	3, 5	pm3.2i 471	. . 3 |- ( ( ch \/_ ze ) /\ ( th \/_ ze ) )
7		mdandyvrx4.2	. . . 4 |- ( ps \/_ si )
8		mdandyvrx4.5	. . . 4 |- ( ta <-> ps )
9	7, 8	axorbciffatcxorb 41072	. . 3 |- ( ta \/_ si )
10	6, 9	pm3.2i 471	. 2 |- ( ( ( ch \/_ ze ) /\ ( th \/_ ze ) ) /\ ( ta \/_ si ) )
11		mdandyvrx4.6	. . 3 |- ( et <-> ph )
12	1, 11	axorbciffatcxorb 41072	. 2 |- ( et \/_ ze )
13	10, 12	pm3.2i 471	1 |- ( ( ( ( ch \/_ ze ) /\ ( th \/_ ze ) ) /\ ( ta \/_ si ) ) /\ ( 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:  mdandyvrx11  41159