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Theorem mpteq1df 41865
 Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
mpteq1df.1 𝑥𝜑
mpteq1df.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
mpteq1df (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Proof of Theorem mpteq1df
StepHypRef Expression
1 mpteq1df.1 . . 3 𝑥𝜑
2 mpteq1df.2 . . 3 (𝜑𝐴 = 𝐵)
31, 2alrimi 2212 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐵)
4 eqid 2801 . . 3 𝐶 = 𝐶
54rgenw 3121 . 2 𝑥𝐴 𝐶 = 𝐶
6 mpteq12f 5116 . 2 ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
73, 5, 6sylancl 589 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538  Ⅎwnf 1785  ∀wral 3109   ↦ cmpt 5113 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773 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 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-opab 5096  df-mpt 5114 This theorem is referenced by:  smfliminflem  43454
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