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Theorem nfrecs 7034
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 7032 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
2 nfcv 2622 . . . . 5  |-  F/_ x On
3 nfv 1678 . . . . . 6  |-  F/ x  a  Fn  b
4 nfcv 2622 . . . . . . 7  |-  F/_ x
b
5 nfrecs.f . . . . . . . . 9  |-  F/_ x F
6 nfcv 2622 . . . . . . . . 9  |-  F/_ x
( a  |`  c
)
75, 6nffv 5864 . . . . . . . 8  |-  F/_ x
( F `  (
a  |`  c ) )
87nfeq2 2639 . . . . . . 7  |-  F/ x
( a `  c
)  =  ( F `
 ( a  |`  c ) )
94, 8nfral 2843 . . . . . 6  |-  F/ x A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )
103, 9nfan 1870 . . . . 5  |-  F/ x
( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
112, 10nfrex 2920 . . . 4  |-  F/ x E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
1211nfab 2626 . . 3  |-  F/_ x { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
1312nfuni 4244 . 2  |-  F/_ x U. { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
141, 13nfcxfr 2620 1  |-  F/_ xrecs ( F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374   {cab 2445   F/_wnfc 2608   A.wral 2807   E.wrex 2808   U.cuni 4238   Oncon0 4871    |` cres 4994    Fn wfn 5574   ` cfv 5579  recscrecs 7031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-recs 7032
This theorem is referenced by:  nfrdg  7070  nfoi  7928  aomclem8  30600
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