Theorem mdandyv12

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	mdandyv12.1	|- ( ph <-> F. )
	mdandyv12.2	|- ( ps <-> T. )
	mdandyv12.3	|- ( ch <-> F. )
	mdandyv12.4	|- ( th <-> F. )
	mdandyv12.5	|- ( ta <-> T. )
	mdandyv12.6	|- ( et <-> T. )
Assertion	mdandyv12	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		mdandyv12.1	|- ( ph <-> F. )
2		mdandyv12.2	|- ( ps <-> T. )
3		mdandyv12.3	|- ( ch <-> F. )
4		mdandyv12.4	|- ( th <-> F. )
5		mdandyv12.5	|- ( ta <-> T. )
6		mdandyv12.6	|- ( et <-> T. )
7	3 1	bothfbothsame	|- ( ch <-> ph )
8	4 1	bothfbothsame	|- ( th <-> ph )
9	7 8	pm3.2i	|- ( ( ch <-> ph ) /\ ( th <-> ph ) )
10	5 2	bothtbothsame	|- ( ta <-> ps )
11	9 10	pm3.2i	|- ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) )
12	6 2	bothtbothsame	|- ( et <-> ps )
13	11 12	pm3.2i	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ps ) )