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Theorem psseq12d 4068
Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
Hypotheses
Ref Expression
psseq1d.1 (𝜑𝐴 = 𝐵)
psseq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
psseq12d (𝜑 → (𝐴𝐶𝐵𝐷))

Proof of Theorem psseq12d
StepHypRef Expression
1 psseq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21psseq1d 4066 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 psseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43psseq2d 4067 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 280 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  fin23lem32  9754  fin23lem34  9756  fin23lem35  9757  fin23lem41  9762  isf32lem5  9767  isf32lem6  9768  isf32lem11  9773  compssiso  9784  chnle  29218
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