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Theorem reseq12d 4640
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 4638 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 reseqd.2 . . 3 (𝜑𝐶 = 𝐷)
43reseq2d 4639 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2088 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cres 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-opab 3846  df-xp 4378  df-res 4384
This theorem is referenced by:  tfrlem3ag  5954  tfrlem3a  5955  tfrlemi1  5976
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