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Theorem mpteq12df 5160
Description: An equality inference for the maps-to notation. Compare mpteq12dv 5165. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.)
Hypotheses
Ref Expression
mpteq12df.1 𝑥𝜑
mpteq12df.2 (𝜑𝐴 = 𝐶)
mpteq12df.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12df (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12df
StepHypRef Expression
1 mpteq12df.1 . 2 𝑥𝜑
2 mpteq12df.2 . 2 (𝜑𝐴 = 𝐶)
3 mpteq12df.3 . . 3 (𝜑𝐵 = 𝐷)
43adantr 481 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
51, 2, 4mpteq12da 5159 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wnf 1786  wcel 2106  cmpt 5157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-opab 5137  df-mpt 5158
This theorem is referenced by:  esumrnmpt2  32036  exrecfnlem  35550  mpteq1df  42779  smflimmpt  44343
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