Theorem mdandyv4

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

Proof

Step	Hyp	Ref	Expression
1		mdandyv4.1	|- ( ph <-> F. )
2		mdandyv4.2	|- ( ps <-> T. )
3		mdandyv4.3	|- ( ch <-> F. )
4		mdandyv4.4	|- ( th <-> F. )
5		mdandyv4.5	|- ( ta <-> T. )
6		mdandyv4.6	|- ( et <-> F. )
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 1	bothfbothsame	|- ( et <-> ph )
13	11 12	pm3.2i	|- ( ( ( ( ch <-> ph ) /\ ( th <-> ph ) ) /\ ( ta <-> ps ) ) /\ ( et <-> ph ) )