Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpteq12dv GIF version

Theorem mpteq12dv 4019
 Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12dv.1 (𝜑𝐴 = 𝐶)
mpteq12dv.2 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12dv (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12dv
StepHypRef Expression
1 mpteq12dv.1 . 2 (𝜑𝐴 = 𝐶)
2 mpteq12dv.2 . . 3 (𝜑𝐵 = 𝐷)
32adantr 274 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
41, 3mpteq12dva 4018 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481   ↦ cmpt 3998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-opab 3999  df-mpt 4000 This theorem is referenced by:  mpteq12i  4025  offval  6000  offval3  6043  restval  12199  ntrfval  12342  clsfval  12343  neifval  12382  cnpfval  12437  cnprcl2k  12448  reldvg  12890  dvfvalap  12892  eldvap  12893
 Copyright terms: Public domain W3C validator