Theorem mdandyvr1 41133

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

Hypotheses

Ref	Expression
mdandyvr1.1	|- ( ph <-> ze )
mdandyvr1.2	|- ( ps <-> si )
mdandyvr1.3	|- ( ch <-> ps )
mdandyvr1.4	|- ( th <-> ph )
mdandyvr1.5	|- ( ta <-> ph )
mdandyvr1.6	|- ( et <-> ph )

Assertion

Ref	Expression
mdandyvr1	|- ( ( ( ( ch <-> si ) /\ ( th <-> ze ) ) /\ ( ta <-> ze ) ) /\ ( et <-> ze ) )

Proof of Theorem mdandyvr1

Step	Hyp	Ref	Expression
1		mdandyvr1.3	. . . . 5 |- ( ch <-> ps )
2		mdandyvr1.2	. . . . 5 |- ( ps <-> si )
3	1, 2	bitri 264	. . . 4 |- ( ch <-> si )
4		mdandyvr1.4	. . . . 5 |- ( th <-> ph )
5		mdandyvr1.1	. . . . 5 |- ( ph <-> ze )
6	4, 5	bitri 264	. . . 4 |- ( th <-> ze )
7	3, 6	pm3.2i 471	. . 3 |- ( ( ch <-> si ) /\ ( th <-> ze ) )
8		mdandyvr1.5	. . . 4 |- ( ta <-> ph )
9	8, 5	bitri 264	. . 3 |- ( ta <-> ze )
10	7, 9	pm3.2i 471	. 2 |- ( ( ( ch <-> si ) /\ ( th <-> ze ) ) /\ ( ta <-> ze ) )
11		mdandyvr1.6	. . 3 |- ( et <-> ph )
12	11, 5	bitri 264	. 2 |- ( et <-> ze )
13	10, 12	pm3.2i 471	1 |- ( ( ( ( ch <-> si ) /\ ( th <-> ze ) ) /\ ( ta <-> ze ) ) /\ ( et <-> ze ) )

Syntax hints:   <-> wb 196   /\ wa 384

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

This theorem is referenced by:  mdandyvr14  41146