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Theorem nfmpt2 5515
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt2.1 zA
nfmpt2.2 zB
nfmpt2.3 z𝐶
Assertion
Ref Expression
nfmpt2 z(x A, y B𝐶)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   A(x,y,z)   B(x,y,z)   𝐶(x,y,z)

Proof of Theorem nfmpt2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 df-mpt2 5460 . 2 (x A, y B𝐶) = {⟨⟨x, y⟩, w⟩ ∣ ((x A y B) w = 𝐶)}
2 nfmpt2.1 . . . . . 6 zA
32nfcri 2169 . . . . 5 z x A
4 nfmpt2.2 . . . . . 6 zB
54nfcri 2169 . . . . 5 z y B
63, 5nfan 1454 . . . 4 z(x A y B)
7 nfmpt2.3 . . . . 5 z𝐶
87nfeq2 2186 . . . 4 z w = 𝐶
96, 8nfan 1454 . . 3 z((x A y B) w = 𝐶)
109nfoprab 5499 . 2 z{⟨⟨x, y⟩, w⟩ ∣ ((x A y B) w = 𝐶)}
111, 10nfcxfr 2172 1 z(x A, y B𝐶)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  wnfc 2162  {coprab 5456  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-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-oprab 5459  df-mpt2 5460
This theorem is referenced by:  nfiseq  8898
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