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Theorem breqan12d 3882
Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypotheses
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
breqan12i.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
breqan12d ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breqan12d
StepHypRef Expression
1 breq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 breqan12i.2 . 2 (𝜓𝐶 = 𝐷)
3 breq12 3872 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3syl2an 284 1 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296   class class class wbr 3867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868
This theorem is referenced by:  breqan12rd  3883  sosng  4540  isoresbr  5626  isoid  5627  isores3  5632  isoini2  5636  ofrfval  5902  oviec  6438  enqbreq2  7013  ltresr2  7474  axpre-ltadd  7518  leltadd  8022  xltneg  9402  lt2sq  10159  le2sq  10160  sqrtle  10600  sqrtlt  10601  absext  10627
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