New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  psseq2 GIF version

Theorem psseq2 3357
 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 3293 . . 3 (A = B → (C AC B))
2 neeq2 2525 . . 3 (A = B → (CACB))
31, 2anbi12d 691 . 2 (A = B → ((C A CA) ↔ (C B CB)))
4 df-pss 3261 . 2 (CA ↔ (C A CA))
5 df-pss 3261 . 2 (CB ↔ (C B CB))
63, 4, 53bitr4g 279 1 (A = B → (CACB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  psseq2i  3359  psseq2d  3362  psssstr  3375
 Copyright terms: Public domain W3C validator