Theorem psseq12d

Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004)

	Ref	Expression
Hypotheses	psseq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	psseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	psseq12d	⊢ ( 𝜑 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		psseq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		psseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	psseq1d	⊢ ( 𝜑 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )
4	2	psseq2d	⊢ ( 𝜑 → ( 𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷 ) )