Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq12i Structured version   Visualization version   GIF version

Theorem psseq12i 4054
 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 (𝐴𝐶𝐵𝐷)

Proof of Theorem psseq12i
StepHypRef Expression
1 psseq1i.1 . . 3 𝐴 = 𝐵
21psseq1i 4052 . 2 (𝐴𝐶𝐵𝐶)
3 psseq12i.2 . . 3 𝐶 = 𝐷
43psseq2i 4053 . 2 (𝐵𝐶𝐵𝐷)
52, 4bitri 278 1 (𝐴𝐶𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ⊊ wpss 3920 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-v 3482  df-in 3926  df-ss 3936  df-pss 3938 This theorem is referenced by:  canthp1lem2  10073  symgvalstruct  18525
 Copyright terms: Public domain W3C validator