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Theorem nfrecs 6635
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 6633 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
2 nfcv 2572 . . . . 5  |-  F/_ x On
3 nfv 1629 . . . . . 6  |-  F/ x  a  Fn  b
4 nfcv 2572 . . . . . . 7  |-  F/_ x
b
5 nfrecs.f . . . . . . . . 9  |-  F/_ x F
6 nfcv 2572 . . . . . . . . 9  |-  F/_ x
( a  |`  c
)
75, 6nffv 5735 . . . . . . . 8  |-  F/_ x
( F `  (
a  |`  c ) )
87nfeq2 2583 . . . . . . 7  |-  F/ x
( a `  c
)  =  ( F `
 ( a  |`  c ) )
94, 8nfral 2759 . . . . . 6  |-  F/ x A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )
103, 9nfan 1846 . . . . 5  |-  F/ x
( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
112, 10nfrex 2761 . . . 4  |-  F/ x E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
1211nfab 2576 . . 3  |-  F/_ x { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
1312nfuni 4021 . 2  |-  F/_ x U. { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
141, 13nfcxfr 2569 1  |-  F/_ xrecs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   {cab 2422   F/_wnfc 2559   A.wral 2705   E.wrex 2706   U.cuni 4015   Oncon0 4581    |` cres 4880    Fn wfn 5449   ` cfv 5454  recscrecs 6632
This theorem is referenced by:  nfrdg  6672  nfoi  7483  aomclem8  27136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-recs 6633
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