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Theorem psseq12d 4029
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 4027 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 psseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43psseq2d 4028 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 278 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wpss 3888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-in 3894  df-ss 3904  df-pss 3906
This theorem is referenced by:  fin23lem32  10100  fin23lem34  10102  fin23lem35  10103  fin23lem41  10108  isf32lem5  10113  isf32lem6  10114  isf32lem11  10119  compssiso  10130  chnle  29876
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