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Theorem mpteq12dv 4197
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 4196 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1398  wcel 2205  cmpt 4176
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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216
This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-opab 4177  df-mpt 4178
This theorem is referenced by:  mpteq12i  4203  offval  6283  offval3  6340  ccatfvalfi  11305  swrdval  11365  odzval  12964  restval  13542  prdsex  13566  prdsval  13570  qusval  13620  grpinvfvalg  13839  grpinvpropdg  13872  opprnegg  14312  lspfval  14648  lsppropd  14692  sraval  14697  psrval  14926  ntrfval  15077  clsfval  15078  neifval  15117  cnpfval  15172  cnprcl2k  15183  reldvg  15656  dvfvalap  15658  eldvap  15659  vtxdgfval  16395  vtxdgop  16399  vtxdeqd  16403
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