Theorem rr-phpd

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

	Ref	Expression
Hypotheses	rr-phpd.1	$$⊢\phi \to A\in \mathrm{\omega }$$
	rr-phpd.2	$$⊢\phi \to B\subseteq A$$
	rr-phpd.3	$$⊢\phi \to A\approx B$$
Assertion	rr-phpd	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		rr-phpd.1	$$⊢\phi \to A\in \mathrm{\omega }$$
2		rr-phpd.2	$$⊢\phi \to B\subseteq A$$
3		rr-phpd.3	$$⊢\phi \to A\approx B$$
4	2	adantr	$$⊢\left(\phi \wedge \neg A=B\right)\to B\subseteq A$$
5		simpr	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg A=B$$
6	5	neqcomd	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg B=A$$
7		dfpss2	$$⊢B\subset A\leftrightarrow \left(B\subseteq A\wedge \neg B=A\right)$$
8	4 6 7	sylanbrc	$$⊢\left(\phi \wedge \neg A=B\right)\to B\subset A$$
9		php	$$⊢\left(A\in \mathrm{\omega }\wedge B\subset A\right)\to \neg A\approx B$$
10	1 8 9	syl2an2r	$$⊢\left(\phi \wedge \neg A=B\right)\to \neg A\approx B$$
11	10	ex	$$⊢\phi \to \left(\neg A=B\to \neg A\approx B\right)$$
12	3 11	mt4d	$$⊢\phi \to A=B$$