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Theorem nfrecs 5953
Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
nfrecs.f  |-  F/_ x F
Assertion
Ref Expression
nfrecs  |-  F/_ xrecs ( F )

Proof of Theorem nfrecs
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-recs 5951 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
2 nfcv 2194 . . . . 5  |-  F/_ x On
3 nfv 1437 . . . . . 6  |-  F/ x  a  Fn  b
4 nfcv 2194 . . . . . . 7  |-  F/_ x
b
5 nfrecs.f . . . . . . . . 9  |-  F/_ x F
6 nfcv 2194 . . . . . . . . 9  |-  F/_ x
( a  |`  c
)
75, 6nffv 5213 . . . . . . . 8  |-  F/_ x
( F `  (
a  |`  c ) )
87nfeq2 2205 . . . . . . 7  |-  F/ x
( a `  c
)  =  ( F `
 ( a  |`  c ) )
94, 8nfralxy 2377 . . . . . 6  |-  F/ x A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )
103, 9nfan 1473 . . . . 5  |-  F/ x
( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
112, 10nfrexxy 2378 . . . 4  |-  F/ x E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
1211nfab 2198 . . 3  |-  F/_ x { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
1312nfuni 3614 . 2  |-  F/_ x U. { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
141, 13nfcxfr 2191 1  |-  F/_ xrecs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   {cab 2042   F/_wnfc 2181   A.wral 2323   E.wrex 2324   U.cuni 3608   Oncon0 4128    |` cres 4375    Fn wfn 4925   ` cfv 4930  recscrecs 5950
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-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-recs 5951
This theorem is referenced by:  nffrec  6013
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