Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpteq1i Structured version   Visualization version   GIF version

Theorem mpteq1i 4667
 Description: An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
mpteq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
mpteq1i (𝑥𝐴𝐶) = (𝑥𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1i
StepHypRef Expression
1 mpteq1i.1 . 2 𝐴 = 𝐵
2 mpteq1 4665 . 2 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
31, 2ax-mp 5 1 (𝑥𝐴𝐶) = (𝑥𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ↦ cmpt 4643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-opab 4644  df-mpt 4645 This theorem is referenced by:  sge0iunmptlemfi  39306  sge0iunmpt  39311  hoidmvlelem3  39487  smfmulc1  39681  wlknwwlksnbij2  41089  wlkwwlkbij2  41096  wwlksnextbij  41108
 Copyright terms: Public domain W3C validator