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Theorem reseq12d 4926
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 4924 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 reseqd.2 . . 3 (𝜑𝐶 = 𝐷)
43reseq2d 4925 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2222 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cres 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-opab 4080  df-xp 4650  df-res 4656
This theorem is referenced by:  tfrlem3ag  6333  tfrlem3a  6334  tfrlemi1  6356  tfr1onlem3ag  6361  setsvalg  12541  znval  13929  psrval  13941  isxms  14403  isms  14405
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