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Theorem mpteq1df 39265
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 2081 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐵)
4 eqid 2621 . . . 4 𝐶 = 𝐶
54rgenw 2923 . . 3 𝑥𝐴 𝐶 = 𝐶
65a1i 11 . 2 (𝜑 → ∀𝑥𝐴 𝐶 = 𝐶)
7 mpteq12f 4729 . 2 ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
83, 6, 7syl2anc 693 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480   = wceq 1482  wnf 1707  wral 2911  cmpt 4727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2916  df-opab 4711  df-mpt 4728
This theorem is referenced by:  smfliminflem  40805
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