ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpteq2ia GIF version

Theorem mpteq2ia 3954
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpteq2ia.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2ia (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Proof of Theorem mpteq2ia
StepHypRef Expression
1 eqid 2100 . . 3 𝐴 = 𝐴
21ax-gen 1393 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 2444 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 3948 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 420 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1297   = wceq 1299  wcel 1448  wral 2375  cmpt 3929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-opab 3930  df-mpt 3931
This theorem is referenced by:  mpteq2i  3955  feqresmpt  5407  elfvmptrab  5448  fmptap  5542  offres  5964  cnrecnv  10523  ege2le3  11175  eirraplem  11278  cnmpt1st  12238  cnmpt2nd  12239  expcncf  12504
  Copyright terms: Public domain W3C validator