Theorem mdandyv14

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	mdandyv14.1	|- ( ph <-> F. )
	mdandyv14.2	|- ( ps <-> T. )
	mdandyv14.3	|- ( ch <-> F. )
	mdandyv14.4	|- ( th <-> T. )
	mdandyv14.5	|- ( ta <-> T. )
	mdandyv14.6	|- ( et <-> T. )
Assertion	mdandyv14	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ps ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		mdandyv14.1	|- ( ph <-> F. )
2		mdandyv14.2	|- ( ps <-> T. )
3		mdandyv14.3	|- ( ch <-> F. )
4		mdandyv14.4	|- ( th <-> T. )
5		mdandyv14.5	|- ( ta <-> T. )
6		mdandyv14.6	|- ( et <-> T. )
7	3 1	bothfbothsame	|- ( ch <-> ph )
8	4 2	bothtbothsame	|- ( th <-> ps )
9	7 8	pm3.2i	|- ( ( ch <-> ph ) /\ ( th <-> ps ) )
10	5 2	bothtbothsame	|- ( ta <-> ps )
11	9 10	pm3.2i	|- ( ( ( ch <-> ph ) /\ ( th <-> ps ) ) /\ ( ta <-> ps ) )
12	6 2	bothtbothsame	|- ( et <-> ps )
13	11 12	pm3.2i	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ps ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ps ) )