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Theorem psseq12d 3685
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 3683 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 psseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43psseq2d 3684 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 268 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wpss 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-in 3567  df-ss 3574  df-pss 3576
This theorem is referenced by:  fin23lem32  9126  fin23lem34  9128  fin23lem35  9129  fin23lem41  9134  isf32lem5  9139  isf32lem6  9140  isf32lem11  9145  compssiso  9156  canthp1lem2  9435  chnle  28261
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