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Theorem mpteq1df 40373
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 2199 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐵)
4 eqid 2778 . . . 4 𝐶 = 𝐶
54rgenw 3106 . . 3 𝑥𝐴 𝐶 = 𝐶
65a1i 11 . 2 (𝜑 → ∀𝑥𝐴 𝐶 = 𝐶)
7 mpteq12f 4969 . 2 ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
83, 6, 7syl2anc 579 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1599   = wceq 1601  wnf 1827  wral 3090  cmpt 4967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-opab 4951  df-mpt 4968
This theorem is referenced by:  smfliminflem  41977
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