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Theorem xrneq12d 37767
Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
Hypotheses
Ref Expression
xrneq12d.1 (𝜑𝐴 = 𝐵)
xrneq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
xrneq12d (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem xrneq12d
StepHypRef Expression
1 xrneq12d.1 . 2 (𝜑𝐴 = 𝐵)
2 xrneq12d.2 . 2 (𝜑𝐶 = 𝐷)
3 xrneq12 37765 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2anc 583 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cxrn 37554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204  df-co 5678  df-xrn 37753
This theorem is referenced by: (None)
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