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Theorem mpt2eq123dv 5595
 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 265 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
4 mpt2eq123dv.3 . . 3 (𝜑𝐶 = 𝐹)
54adantr 265 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
61, 3, 5mpt2eq123dva 5594 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409   ↦ cmpt2 5542 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-oprab 5544  df-mpt2 5545 This theorem is referenced by:  mpt2eq123i  5596
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