Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  psseq2d Structured version   Visualization version   GIF version

Theorem psseq2d 3662
 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
psseq2d (𝜑 → (𝐶𝐴𝐶𝐵))

Proof of Theorem psseq2d
StepHypRef Expression
1 psseq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 psseq2 3657 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2syl 17 1 (𝜑 → (𝐶𝐴𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  psseq12d  3663  php3  8031  inf3lem5  8412  infeq5i  8416  ackbij1lem15  8939  fin4en1  9014  chpsscon1  27747  chnle  27757  atcvatlem  28628  atcvati  28629  lsatcvat  33355
 Copyright terms: Public domain W3C validator