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Theorem nfres 4614
 Description: Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
nfres.1 𝑥𝐴
nfres.2 𝑥𝐵
Assertion
Ref Expression
nfres 𝑥(𝐴𝐵)

Proof of Theorem nfres
StepHypRef Expression
1 df-res 4357 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2178 . . . 4 𝑥V
53, 4nfxp 4371 . . 3 𝑥(𝐵 × V)
62, 5nfin 3143 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2175 1 𝑥(𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  Ⅎwnfc 2165  Vcvv 2557   ∩ cin 2916   × cxp 4343   ↾ cres 4347 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924  df-opab 3819  df-xp 4351  df-res 4357 This theorem is referenced by:  nfima  4676  nffrec  5982
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