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Theorem nffr 4114
Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nffr  |-  F/ x  R  Fr  A

Proof of Theorem nffr
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 df-frind 4097 . 2  |-  ( R  Fr  A  <->  A. sFrFor  R A s )
2 nffr.r . . . 4  |-  F/_ x R
3 nffr.a . . . 4  |-  F/_ x A
4 nfcv 2194 . . . 4  |-  F/_ x
s
52, 3, 4nffrfor 4113 . . 3  |-  F/ xFrFor  R A s
65nfal 1484 . 2  |-  F/ x A. sFrFor  R A s
71, 6nfxfr 1379 1  |-  F/ x  R  Fr  A
Colors of variables: wff set class
Syntax hints:   A.wal 1257   F/wnf 1365   F/_wnfc 2181  FrFor wfrfor 4092    Fr wfr 4093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-frfor 4096  df-frind 4097
This theorem is referenced by:  nfwe  4120
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