Theorem psseq12i 4028

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 4026	. 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
3		psseq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	psseq2i 4027	. 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
5	2, 4	bitri 277	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)

Syntax hints:   ↔ wb 208   = wceq 1548   ⊊ wpss 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-ne 2937  df-ss 3902  df-pss 3905

This theorem is referenced by:  canthp1lem2  10571  symgvalstruct  19367