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Theorem difeq12d 3057
Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
Hypotheses
Ref Expression
difeq12d.1 (φA = B)
difeq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
difeq12d (φ → (A𝐶) = (B𝐷))

Proof of Theorem difeq12d
StepHypRef Expression
1 difeq12d.1 . . 3 (φA = B)
21difeq1d 3055 . 2 (φ → (A𝐶) = (B𝐶))
3 difeq12d.2 . . 3 (φ𝐶 = 𝐷)
43difeq2d 3056 . 2 (φ → (B𝐶) = (B𝐷))
52, 4eqtrd 2069 1 (φ → (A𝐶) = (B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cdif 2908
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-in1 544  ax-in2 545  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-bnd 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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-dif 2914
This theorem is referenced by: (None)
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