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Theorem xrneq12 35515
Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
Assertion
Ref Expression
xrneq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem xrneq12
StepHypRef Expression
1 xrneq1 35509 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 xrneq2 35512 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2873 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  cxrn 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-in 3940  df-ss 3949  df-br 5058  df-opab 5120  df-co 5557  df-xrn 35503
This theorem is referenced by:  xrneq12i  35516  xrneq12d  35517
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