Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq2i Structured version   Visualization version   GIF version

Theorem psseq2i 4067
 Description: An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypothesis
Ref Expression
psseq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
psseq2i (𝐶𝐴𝐶𝐵)

Proof of Theorem psseq2i
StepHypRef Expression
1 psseq1i.1 . 2 𝐴 = 𝐵
2 psseq2 4065 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2ax-mp 5 1 (𝐶𝐴𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1533   ⊊ wpss 3937 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 2156  ax-12 2172  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 3943  df-ss 3952  df-pss 3954 This theorem is referenced by:  psseq12i  4068  disjpss  4410  infeq5i  9093
 Copyright terms: Public domain W3C validator