Theorem psseq12i 4067

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 4065	. 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
3		psseq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	psseq2i 4066	. 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
5	2, 4	bitri 277	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)

Syntax hints:   ↔ wb 208   = wceq 1533   ⊊ wpss 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3942  df-ss 3951  df-pss 3953

This theorem is referenced by:  canthp1lem2  10069  symgvalstruct  18519