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Theorem imaeq12d 4612
 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 4610 . 2 (φ → (A𝐶) = (B𝐶))
3 imaeq12d.2 . . 3 (φ𝐶 = 𝐷)
43imaeq2d 4611 . 2 (φ → (B𝐶) = (B𝐷))
52, 4eqtrd 2069 1 (φ → (A𝐶) = (B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   “ cima 4291 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  csbima12g  4629
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