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Theorem sbcrext 2986
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext  |-  ( F/_ y A  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem sbcrext
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcex 2917 . . 3  |-  ( [. A  /  x ]. E. y  e.  B  ph  ->  A  e.  _V )
21a1i 9 . 2  |-  ( F/_ y A  ->  ( [. A  /  x ]. E. y  e.  B  ph  ->  A  e.  _V ) )
3 nfnfc1 2284 . . 3  |-  F/ y
F/_ y A
4 id 19 . . . 4  |-  ( F/_ y A  ->  F/_ y A )
5 nfcvd 2282 . . . 4  |-  ( F/_ y A  ->  F/_ y _V )
64, 5nfeld 2297 . . 3  |-  ( F/_ y A  ->  F/ y  A  e.  _V )
7 sbcex 2917 . . . 4  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
872a1i 27 . . 3  |-  ( F/_ y A  ->  ( y  e.  B  ->  ( [. A  /  x ]. ph  ->  A  e.  _V ) ) )
93, 6, 8rexlimd2 2547 . 2  |-  ( F/_ y A  ->  ( E. y  e.  B  [. A  /  x ]. ph  ->  A  e.  _V ) )
10 sbcco 2930 . . . 4  |-  ( [. A  /  z ]. [. z  /  x ]. E. y  e.  B  ph  <->  [. A  /  x ]. E. y  e.  B  ph )
11 simpl 108 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ y A )  ->  A  e.  _V )
12 sbsbc 2913 . . . . . . 7  |-  ( [ z  /  x ] E. y  e.  B  ph  <->  [. z  /  x ]. E. y  e.  B  ph )
13 nfcv 2281 . . . . . . . . 9  |-  F/_ x B
14 nfs1v 1912 . . . . . . . . 9  |-  F/ x [ z  /  x ] ph
1513, 14nfrexxy 2472 . . . . . . . 8  |-  F/ x E. y  e.  B  [ z  /  x ] ph
16 sbequ12 1744 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
1716rexbidv 2438 . . . . . . . 8  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph ) )
1815, 17sbie 1764 . . . . . . 7  |-  ( [ z  /  x ] E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph )
1912, 18bitr3i 185 . . . . . 6  |-  ( [. z  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [ z  /  x ] ph )
20 nfcvd 2282 . . . . . . . . . 10  |-  ( F/_ y A  ->  F/_ y
z )
2120, 4nfeqd 2296 . . . . . . . . 9  |-  ( F/_ y A  ->  F/ y  z  =  A )
223, 21nfan1 1543 . . . . . . . 8  |-  F/ y ( F/_ y A  /\  z  =  A )
23 dfsbcq2 2912 . . . . . . . . 9  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
2423adantl 275 . . . . . . . 8  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
2522, 24rexbid 2436 . . . . . . 7  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( E. y  e.  B  [ z  /  x ] ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
2625adantll 467 . . . . . 6  |-  ( ( ( A  e.  _V  /\ 
F/_ y A )  /\  z  =  A )  ->  ( E. y  e.  B  [
z  /  x ] ph 
<->  E. y  e.  B  [. A  /  x ]. ph ) )
2719, 26syl5bb 191 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ y A )  /\  z  =  A )  ->  ( [. z  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
2811, 27sbcied 2945 . . . 4  |-  ( ( A  e.  _V  /\  F/_ y A )  -> 
( [. A  /  z ]. [. z  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
)
2910, 28bitr3id 193 . . 3  |-  ( ( A  e.  _V  /\  F/_ y A )  -> 
( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
)
3029expcom 115 . 2  |-  ( F/_ y A  ->  ( A  e.  _V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
) )
312, 9, 30pm5.21ndd 694 1  |-  ( F/_ y A  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   [wsb 1735   F/_wnfc 2268   E.wrex 2417   _Vcvv 2686   [.wsbc 2909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910
This theorem is referenced by:  sbcrex  2988
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