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Theorem xrneq12 38384
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 38378 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 xrneq2 38381 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2797 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-in 3958  df-ss 3968  df-br 5144  df-opab 5206  df-co 5694  df-xrn 38372
This theorem is referenced by:  xrneq12i  38385  xrneq12d  38386
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