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Theorem mpteq12dv 3950
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 272 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
41, 3mpteq12dva 3949 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  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:  mpteq12i  3956  offval  5921  offval3  5963  restval  11908  ntrfval  12051  clsfval  12052  neifval  12091  cnpfval  12146  cnprcl2k  12156  reldvg  12521  dvfvalap  12523  eldvap  12524
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