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Theorem psseq12d 4071
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 4069 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 psseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43psseq2d 4070 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 281 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  wpss 3937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3943  df-ss 3952  df-pss 3954
This theorem is referenced by:  fin23lem32  9760  fin23lem34  9762  fin23lem35  9763  fin23lem41  9768  isf32lem5  9773  isf32lem6  9774  isf32lem11  9779  compssiso  9790  chnle  29285
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