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Theorem reseq12d 4556
 Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
reseqd.1 (φA = B)
reseqd.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
reseq12d (φ → (A𝐶) = (B𝐷))

Proof of Theorem reseq12d
StepHypRef Expression
1 reseqd.1 . . 3 (φA = B)
21reseq1d 4554 . 2 (φ → (A𝐶) = (B𝐶))
3 reseqd.2 . . 3 (φ𝐶 = 𝐷)
43reseq2d 4555 . 2 (φ → (B𝐶) = (B𝐷))
52, 4eqtrd 2069 1 (φ → (A𝐶) = (B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ↾ cres 4290 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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-opab 3810  df-xp 4294  df-res 4300 This theorem is referenced by:  tfrlem3ag  5865  tfrlem3a  5866  tfrlemi1  5887
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