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Theorem mpteq1d 3973
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
mpteq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
mpteq1d (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem mpteq1d
StepHypRef Expression
1 mpteq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 mpteq1 3972 . 2 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cmpt 3949
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2395  df-opab 3950  df-mpt 3951
This theorem is referenced by:  fmptapd  5565  offval  5943
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