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Theorem psseq2 3678
Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psseq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem psseq2
StepHypRef Expression
1 sseq2 3611 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
2 neeq2 2853 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2anbi12d 746 . 2 (𝐴 = 𝐵 → ((𝐶𝐴𝐶𝐴) ↔ (𝐶𝐵𝐶𝐵)))
4 df-pss 3575 . 2 (𝐶𝐴 ↔ (𝐶𝐴𝐶𝐴))
5 df-pss 3575 . 2 (𝐶𝐵 ↔ (𝐶𝐵𝐶𝐵))
63, 4, 53bitr4g 303 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wne 2790  wss 3559  wpss 3560
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 3566  df-ss 3573  df-pss 3575
This theorem is referenced by:  psseq2i  3680  psseq2d  3683  psssstr  3696  brrpssg  6899  sorpssint  6907  php  8095  php2  8096  pssnn  8129  isfin4  9070  fin2i2  9091  elnp  9760  elnpi  9761  ltprord  9803  pgpfac1lem1  18401  pgpfac1lem5  18406  lbsextlem4  19089  alexsubALTlem4  21773  spansncv  28379  cvbr  29008  cvcon3  29010  cvnbtwn  29012  cvbr4i  29093  dfon2lem6  31421  dfon2lem7  31422  dfon2lem8  31423  dfon2  31425  lcvbr  33815  lcvnbtwn  33819  lsatcv0  33825  lsat0cv  33827  islshpcv  33847  mapdcv  36456
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