Theorem psseq12i 4019

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

Hypotheses

Ref	Expression
psseq1i.1	⊢ 𝐴 = 𝐵
psseq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
psseq12i	⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)

Proof of Theorem psseq12i

Step	Hyp	Ref	Expression
1		psseq1i.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	psseq1i 4017	. 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
3		psseq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	psseq2i 4018	. 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
5	2, 4	bitri 278	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)

Syntax hints:   ↔ wb 209   = wceq 1538   ⊊ wpss 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-in 3888  df-ss 3898  df-pss 3900

This theorem is referenced by:  canthp1lem2  10064  symgvalstruct  18517