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Theorem reseq12d 4890
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 4888 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 reseqd.2 . . 3 (𝜑𝐶 = 𝐷)
43reseq2d 4889 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2203 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  cres 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-opab 4049  df-xp 4615  df-res 4621
This theorem is referenced by:  tfrlem3ag  6285  tfrlem3a  6286  tfrlemi1  6308  tfr1onlem3ag  6313  setsvalg  12433  isxms  13204  isms  13206
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