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Theorem mpt2eq123dv 5509
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 → (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹))
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 ((φ x A) → B = 𝐸)
4 mpt2eq123dv.3 . . 3 (φ𝐶 = 𝐹)
54adantr 261 . 2 ((φ (x A y B)) → 𝐶 = 𝐹)
61, 3, 5mpt2eq123dva 5508 1 (φ → (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cmpt2 5457
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  mpt2eq123i  5510
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