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Theorem imaeq1 4937
 Description: Equality theorem for image. (Contributed by set.mm contributors, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1 (A = B → (AC) = (BC))

Proof of Theorem imaeq1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4641 . . . 4 (A = B → (yAxyBx))
21rexbidv 2635 . . 3 (A = B → (y C yAxy C yBx))
32abbidv 2467 . 2 (A = B → {x y C yAx} = {x y C yBx})
4 df-ima 4727 . 2 (AC) = {x y C yAx}
5 df-ima 4727 . 2 (BC) = {x y C yBx}
63, 4, 53eqtr4g 2410 1 (A = B → (AC) = (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  {cab 2339  ∃wrex 2615   class class class wbr 4639   “ 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620  df-br 4640  df-ima 4727 This theorem is referenced by:  imaeq1i  4939  imaeq1d  4941  rneq  4956  f1imacnv  5302  clos1eq2  5875  eceq2  5963
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