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Theorem mpteq12dv 4100
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 276 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
41, 3mpteq12dva 4099 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  cmpt 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-opab 4080  df-mpt 4081
This theorem is referenced by:  mpteq12i  4106  offval  6113  offval3  6158  odzval  12272  restval  12747  prdsex  12771  qusval  12797  grpinvfvalg  12983  grpinvpropdg  13016  opprnegg  13430  lspfval  13701  lsppropd  13745  sraval  13750  psrval  13941  ntrfval  14052  clsfval  14053  neifval  14092  cnpfval  14147  cnprcl2k  14158  reldvg  14600  dvfvalap  14602  eldvap  14603
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