MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq1d Structured version   Visualization version   GIF version

Theorem psseq1d 3897
Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypothesis
Ref Expression
psseq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
psseq1d (𝜑 → (𝐴𝐶𝐵𝐶))

Proof of Theorem psseq1d
StepHypRef Expression
1 psseq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 psseq1 3892 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
31, 2syl 17 1 (𝜑 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wpss 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-ne 2973  df-in 3777  df-ss 3784  df-pss 3786
This theorem is referenced by:  psseq12d  3899  fin23lem32  9455  fin23lem35  9458  compssiso  9485  mrieqv2d  16613  mrissmrcd  16614  pgpfac1lem5  18793  islbs3  19477  chpsscon2  28888
  Copyright terms: Public domain W3C validator