Theorem rr-phpd

Description: Equivalent of php without negation. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	rr-phpd.1	|- ( ph -> A e. _om )
	rr-phpd.2	|- ( ph -> B C_ A )
	rr-phpd.3	|- ( ph -> A ~~ B )
Assertion	rr-phpd	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		rr-phpd.1	|- ( ph -> A e. _om )
2		rr-phpd.2	|- ( ph -> B C_ A )
3		rr-phpd.3	|- ( ph -> A ~~ B )
4	2	adantr	|- ( ( ph /\ -. A = B ) -> B C_ A )
5		simpr	|- ( ( ph /\ -. A = B ) -> -. A = B )
6	5	neqcomd	|- ( ( ph /\ -. A = B ) -> -. B = A )
7		dfpss2	|- ( B C. A <-> ( B C_ A /\ -. B = A ) )
8	4 6 7	sylanbrc	|- ( ( ph /\ -. A = B ) -> B C. A )
9		php	|- ( ( A e. _om /\ B C. A ) -> -. A ~~ B )
10	1 8 9	syl2an2r	|- ( ( ph /\ -. A = B ) -> -. A ~~ B )
11	10	ex	|- ( ph -> ( -. A = B -> -. A ~~ B ) )
12	3 11	mt4d	|- ( ph -> A = B )