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Theorem mpteq2da 4107
Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq2da.1 𝑥𝜑
mpteq2da.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2da (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2189 . . 3 𝐴 = 𝐴
21ax-gen 1460 . 2 𝑥 𝐴 = 𝐴
3 mpteq2da.1 . . 3 𝑥𝜑
4 mpteq2da.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
54ex 115 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
63, 5ralrimi 2561 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
7 mpteq12f 4098 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
82, 6, 7sylancr 414 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362   = wceq 1364  wnf 1471  wcel 2160  wral 2468  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:  mpteq2dva  4108  prodeq1f  11595  prodeq2  11600
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