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Theorem xrneq12i 37558
Description: Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
Hypotheses
Ref Expression
xrneq12i.1 𝐴 = 𝐵
xrneq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
xrneq12i (𝐴𝐶) = (𝐵𝐷)

Proof of Theorem xrneq12i
StepHypRef Expression
1 xrneq12i.1 . 2 𝐴 = 𝐵
2 xrneq12i.2 . 2 𝐶 = 𝐷
3 xrneq12 37557 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 689 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cxrn 37346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-br 5149  df-opab 5211  df-co 5685  df-xrn 37545
This theorem is referenced by:  xrnres4  37579  xrnresex  37580
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