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Theorem mpt2eq123dv 5567
Description: An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
mpt2eq123dv.1 (𝜑𝐴 = 𝐷)
mpt2eq123dv.2 (𝜑𝐵 = 𝐸)
mpt2eq123dv.3 (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
mpt2eq123dv (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 mpt2eq123dv
StepHypRef Expression
1 mpt2eq123dv.1 . 2 (𝜑𝐴 = 𝐷)
2 mpt2eq123dv.2 . . 3 (𝜑𝐵 = 𝐸)
32adantr 261 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
4 mpt2eq123dv.3 . . 3 (𝜑𝐶 = 𝐹)
54adantr 261 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
61, 3, 5mpt2eq123dva 5566 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-oprab 5516  df-mpt2 5517
This theorem is referenced by:  mpt2eq123i  5568
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