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Theorem mpteq2ia 4023
 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 2140 . . 3 𝐴 = 𝐴
21ax-gen 1426 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 2489 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 4017 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 423 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ↦ 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:  mpteq2i  4024  feqresmpt  5484  elfvmptrab  5525  fmptap  5619  offres  6042  cnrecnv  10734  ege2le3  11434  eirraplem  11539  cnmpt1st  12516  cnmpt2nd  12517  expcncf  12820  dvexp  12903  dveflem  12915  dvef  12916
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