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Theorem mpteq12dv 3886
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 270 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
41, 3mpteq12dva 3885 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  cmpt 3865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-opab 3866  df-mpt 3867
This theorem is referenced by:  mpteq12i  3892  offval  5798  offval3  5840
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