Theorem mdandyv4 41120

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)

Hypotheses

Ref	Expression
mdandyv4.1	|- ( ph <-> F. )
mdandyv4.2	|- ( ps <-> T. )
mdandyv4.3	|- ( ch <-> F. )
mdandyv4.4	|- ( th <-> F. )
mdandyv4.5	|- ( ta <-> T. )
mdandyv4.6	|- ( et <-> F. )

Assertion

Ref	Expression
mdandyv4	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ph ) )

Proof of Theorem mdandyv4

Step	Hyp	Ref	Expression
1		mdandyv4.3	. . . . 5 |- ( ch <-> F. )
2		mdandyv4.1	. . . . 5 |- ( ph <-> F. )
3	1, 2	bothfbothsame 41067	. . . 4 |- ( ch <-> ph )
4		mdandyv4.4	. . . . 5 |- ( th <-> F. )
5	4, 2	bothfbothsame 41067	. . . 4 |- ( th <-> ph )
6	3, 5	pm3.2i 471	. . 3 |- ( ( ch <-> ph ) /\ ( th <-> ph ) )
7		mdandyv4.5	. . . 4 |- ( ta <-> T. )
8		mdandyv4.2	. . . 4 |- ( ps <-> T. )
9	7, 8	bothtbothsame 41066	. . 3 |- ( ta <-> ps )
10	6, 9	pm3.2i 471	. 2 |- ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) )
11		mdandyv4.6	. . 3 |- ( et <-> F. )
12	11, 2	bothfbothsame 41067	. 2 |- ( et <-> ph )
13	10, 12	pm3.2i 471	1 |- ( ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   T. wtru 1484   F. wfal 1488

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: (None)