MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq1i Unicode version

Theorem psseq1i 3265
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
psseq1i  |-  ( A 
C.  C  <->  B  C.  C )

Proof of Theorem psseq1i
StepHypRef Expression
1 psseq1i.1 . 2  |-  A  =  B
2 psseq1 3263 . 2  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
31, 2ax-mp 8 1  |-  ( A 
C.  C  <->  B  C.  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    C. wpss 3153
This theorem is referenced by:  psseq12i  3267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
  Copyright terms: Public domain W3C validator