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Theorem imaeq12d 6014
Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
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 6012 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 imaeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43imaeq2d 6013 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2776 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cima 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646
This theorem is referenced by:  csbima12  6031  predeq123  6254  vdwpc  16852  dmdprd  19777  isunit  20086  qtopval  23046  limciun  25258  ig1pval  25537  ispth  28671  irngval  32359  qqhval  32555  eulerpartgbij  32972  orvcval  33057  ballotlemrval  33117  ballotlemrinv0  33132  ballotlemrinv  33133  mthmval  34169  bj-projeq  35463  itg2addnclem2  36130  islmodfg  41382  heeq12  42038
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