Theorem rr-phpd

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

	Ref	Expression
Hypotheses	rr-phpd.1	⊢ ( 𝜑 → 𝐴 ∈ ω )
	rr-phpd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
	rr-phpd.3	⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
Assertion	rr-phpd	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rr-phpd.1	⊢ ( 𝜑 → 𝐴 ∈ ω )
2		rr-phpd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		rr-phpd.3	⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
4	2	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ⊆ 𝐴 )
5		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 = 𝐵 )
6	5	neqcomd	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐵 = 𝐴 )
7		dfpss2	⊢ ( 𝐵 ⊊ 𝐴 ↔ ( 𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴 ) )
8	4 6 7	sylanbrc	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ⊊ 𝐴 )
9		php	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )
10	1 8 9	syl2an2r	⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 ≈ 𝐵 )
11	10	ex	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → ¬ 𝐴 ≈ 𝐵 ) )
12	3 11	mt4d	⊢ ( 𝜑 → 𝐴 = 𝐵 )