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Theorem nfres 4893
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 4623 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 nfres.1 . . 3 𝑥𝐴
3 nfres.2 . . . 4 𝑥𝐵
4 nfcv 2312 . . . 4 𝑥V
53, 4nfxp 4638 . . 3 𝑥(𝐵 × V)
62, 5nfin 3333 . 2 𝑥(𝐴 ∩ (𝐵 × V))
71, 6nfcxfr 2309 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2299  Vcvv 2730  cin 3120   × cxp 4609  cres 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-in 3127  df-opab 4051  df-xp 4617  df-res 4623
This theorem is referenced by:  nfima  4961  nffrec  6375
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