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Theorem psseq12d 3843
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 3841 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 psseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43psseq2d 3842 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 268 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wpss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ne 2933  df-in 3722  df-ss 3729  df-pss 3731
This theorem is referenced by:  fin23lem32  9358  fin23lem34  9360  fin23lem35  9361  fin23lem41  9366  isf32lem5  9371  isf32lem6  9372  isf32lem11  9377  compssiso  9388  canthp1lem2  9667  chnle  28682
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