New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  imaeq12d GIF version

Theorem imaeq12d 4943
 Description: Equality theorem for image. (Contributed by SF, 8-Jan-2018.)
Hypotheses
Ref Expression
imaeq1d.1 (φA = B)
imaeq12d.2 (φC = D)
Assertion
Ref Expression
imaeq12d (φ → (AC) = (BD))

Proof of Theorem imaeq12d
StepHypRef Expression
1 imaeq1d.1 . . 3 (φA = B)
21imaeq1d 4941 . 2 (φ → (AC) = (BC))
3 imaeq12d.2 . . 3 (φC = D)
43imaeq2d 4942 . 2 (φ → (BC) = (BD))
52, 4eqtrd 2385 1 (φ → (AC) = (BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   “ cima 4722 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-br 4640  df-ima 4727 This theorem is referenced by:  csbima12g  4955
 Copyright terms: Public domain W3C validator