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Theorem breqan12rd 4046
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
breqan12rd ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breqan12rd
StepHypRef Expression
1 breq1d.1 . . 3 (𝜑𝐴 = 𝐵)
2 breqan12i.2 . . 3 (𝜓𝐶 = 𝐷)
31, 2breqan12d 4045 . 2 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
43ancoms 268 1 ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364   class class class wbr 4029
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 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030
This theorem is referenced by:  xltnegi  9891  gausslemma2dlem3  15121
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