Theorem mdandyvr4 41136

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
mdandyvr4.1	|- ( ph <-> ze )
mdandyvr4.2	|- ( ps <-> si )
mdandyvr4.3	|- ( ch <-> ph )
mdandyvr4.4	|- ( th <-> ph )
mdandyvr4.5	|- ( ta <-> ps )
mdandyvr4.6	|- ( et <-> ph )

Assertion

Ref	Expression
mdandyvr4	|- ( ( ( ( ch <-> ze ) /\ ( th <-> ze ) ) /\ ( ta <-> si ) ) /\ ( et <-> ze ) )

Proof of Theorem mdandyvr4

Step	Hyp	Ref	Expression
1		mdandyvr4.3	. . . . 5 |- ( ch <-> ph )
2		mdandyvr4.1	. . . . 5 |- ( ph <-> ze )
3	1, 2	bitri 264	. . . 4 |- ( ch <-> ze )
4		mdandyvr4.4	. . . . 5 |- ( th <-> ph )
5	4, 2	bitri 264	. . . 4 |- ( th <-> ze )
6	3, 5	pm3.2i 471	. . 3 |- ( ( ch <-> ze ) /\ ( th <-> ze ) )
7		mdandyvr4.5	. . . 4 |- ( ta <-> ps )
8		mdandyvr4.2	. . . 4 |- ( ps <-> si )
9	7, 8	bitri 264	. . 3 |- ( ta <-> si )
10	6, 9	pm3.2i 471	. 2 |- ( ( ( ch <-> ze ) /\ ( th <-> ze ) ) /\ ( ta <-> si ) )
11		mdandyvr4.6	. . 3 |- ( et <-> ph )
12	11, 2	bitri 264	. 2 |- ( et <-> ze )
13	10, 12	pm3.2i 471	1 |- ( ( ( ( ch <-> ze ) /\ ( th <-> ze ) ) /\ ( ta <-> si ) ) /\ ( 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:  mdandyvr11  41143