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Theorem mpoeq123dv 5873
 Description: An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
mpoeq123dv.1 (𝜑𝐴 = 𝐷)
mpoeq123dv.2 (𝜑𝐵 = 𝐸)
mpoeq123dv.3 (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
mpoeq123dv (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpoeq123dv
StepHypRef Expression
1 mpoeq123dv.1 . 2 (𝜑𝐴 = 𝐷)
2 mpoeq123dv.2 . . 3 (𝜑𝐵 = 𝐸)
32adantr 274 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
4 mpoeq123dv.3 . . 3 (𝜑𝐶 = 𝐹)
54adantr 274 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
61, 3, 5mpoeq123dva 5872 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125   ∈ cmpo 5816 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-oprab 5818  df-mpo 5819 This theorem is referenced by:  mpoeq123i  5874  blfvalps  12732
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