Theorem mdandyv3

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)

	Ref	Expression
Hypotheses	mdandyv3.1	|- ( ph <-> F. )
	mdandyv3.2	|- ( ps <-> T. )
	mdandyv3.3	|- ( ch <-> T. )
	mdandyv3.4	|- ( th <-> T. )
	mdandyv3.5	|- ( ta <-> F. )
	mdandyv3.6	|- ( et <-> F. )
Assertion	mdandyv3	|- ( ( ( ( ch <-> ps ) /\ ( th <-> ps ) ) /\ ( ta <-> ph ) ) /\ ( et <-> ph ) )

Proof

Step	Hyp	Ref	Expression
1		mdandyv3.1	|- ( ph <-> F. )
2		mdandyv3.2	|- ( ps <-> T. )
3		mdandyv3.3	|- ( ch <-> T. )
4		mdandyv3.4	|- ( th <-> T. )
5		mdandyv3.5	|- ( ta <-> F. )
6		mdandyv3.6	|- ( et <-> F. )
7	3 2	bothtbothsame	|- ( ch <-> ps )
8	4 2	bothtbothsame	|- ( th <-> ps )
9	7 8	pm3.2i	|- ( ( ch <-> ps ) /\ ( th <-> ps ) )
10	5 1	bothfbothsame	|- ( ta <-> ph )
11	9 10	pm3.2i	|- ( ( ( ch <-> ps ) /\ ( th <-> ps ) ) /\ ( ta <-> ph ) )
12	6 1	bothfbothsame	|- ( et <-> ph )
13	11 12	pm3.2i	|- ( ( ( ( ch <-> ps ) /\ ( th <-> ps ) ) /\ ( ta <-> ph ) ) /\ ( et <-> ph ) )