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Theorem psseq2 4062
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 3990 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
2 neeq2 3076 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2anbi12d 630 . 2 (𝐴 = 𝐵 → ((𝐶𝐴𝐶𝐴) ↔ (𝐶𝐵𝐶𝐵)))
4 df-pss 3951 . 2 (𝐶𝐴 ↔ (𝐶𝐴𝐶𝐴))
5 df-pss 3951 . 2 (𝐶𝐵 ↔ (𝐶𝐵𝐶𝐵))
63, 4, 53bitr4g 315 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wne 3013  wss 3933  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  psseq2i  4064  psseq2d  4067  psssstr  4080  brrpssg  7440  sorpssint  7448  php  8689  php2  8690  pssnn  8724  isfin4  9707  fin2i2  9728  elnp  10397  elnpi  10398  ltprord  10440  pgpfac1lem1  19125  pgpfac1lem5  19130  lbsextlem4  19862  alexsubALTlem4  22586  spansncv  29357  cvbr  29986  cvcon3  29988  cvnbtwn  29990  cvbr4i  30071  dfon2lem6  32930  dfon2lem7  32931  dfon2lem8  32932  dfon2  32934  lcvbr  36037  lcvnbtwn  36041  lsatcv0  36047  lsat0cv  36049  islshpcv  36069  mapdcv  38676  pssn0  38991
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