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Theorem xrneq12i 35741
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 35740 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 691 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cxrn 35557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-br 5053  df-opab 5115  df-co 5551  df-xrn 35728
This theorem is referenced by:  xrnres4  35758  xrnresex  35759
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