Theorem psseq12i 4091

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 4089	. 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
3		psseq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	psseq2i 4090	. 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
5	2, 4	bitri 274	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)

Syntax hints:   ↔ wb 205   = wceq 1533   ⊊ wpss 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-in 3956  df-ss 3966  df-pss 3968

This theorem is referenced by:  canthp1lem2  10684  symgvalstruct  19358  symgvalstructOLD  19359