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Theorem nfmpt 3840
 Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt.1 xA
nfmpt.2 xB
Assertion
Ref Expression
nfmpt x(y AB)
Distinct variable group:   x,y
Allowed substitution hints:   A(x,y)   B(x,y)

Proof of Theorem nfmpt
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3811 . 2 (y AB) = {⟨y, z⟩ ∣ (y A z = B)}
2 nfmpt.1 . . . . 5 xA
32nfcri 2169 . . . 4 x y A
4 nfmpt.2 . . . . 5 xB
54nfeq2 2186 . . . 4 x z = B
63, 5nfan 1454 . . 3 x(y A z = B)
76nfopab 3816 . 2 x{⟨y, z⟩ ∣ (y A z = B)}
81, 7nfcxfr 2172 1 x(y AB)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Ⅎwnfc 2162  {copab 3808   ↦ cmpt 3809 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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-nfc 2164  df-opab 3810  df-mpt 3811 This theorem is referenced by:  nffrec  5921
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