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Theorem mpteq1df 44507
Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.)
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 . 2 𝑥𝜑
2 mpteq1df.2 . 2 (𝜑𝐴 = 𝐵)
3 eqidd 2727 . 2 (𝜑𝐶 = 𝐶)
41, 2, 3mpteq12df 5227 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnf 1777  cmpt 5224
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-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-opab 5204  df-mpt 5225
This theorem is referenced by:  smfliminflem  46115
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