 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  coeq1 GIF version

Theorem coeq1 4874
 Description: Equality theorem for composition of two classes. (Contributed by set.mm contributors, 3-Jan-1997.)
Assertion
Ref Expression
coeq1 (A = B → (A C) = (B C))

Proof of Theorem coeq1
StepHypRef Expression
1 coss1 4872 . . 3 (A B → (A C) (B C))
2 coss1 4872 . . 3 (B A → (B C) (A C))
31, 2anim12i 549 . 2 ((A B B A) → ((A C) (B C) (B C) (A C)))
4 eqss 3287 . 2 (A = B ↔ (A B B A))
5 eqss 3287 . 2 ((A C) = (B C) ↔ ((A C) (B C) (B C) (A C)))
63, 4, 53imtr4i 257 1 (A = B → (A C) = (B C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ∘ ccom 4721 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726 This theorem is referenced by:  coeq1i  4876  coeq1d  4878  composevalg  5817  pprodeq1  5834  pprodeq2  5835  enmap2lem3  6065  enmap2lem5  6067
 Copyright terms: Public domain W3C validator