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Theorem strcollnfALT 11527
Description: Alternate proof of strcollnf 11526, not using strcollnft 11525. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
strcollnf.nf  |-  F/ b
ph
Assertion
Ref Expression
strcollnfALT  |-  ( A. x  e.  a  E. y ph  ->  E. b A. y ( y  e.  b  <->  E. x  e.  a 
ph ) )
Distinct variable group:    a, b, x, y
Allowed substitution hints:    ph( x, y, a, b)

Proof of Theorem strcollnfALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 strcoll2 11524 . 2  |-  ( A. x  e.  a  E. y ph  ->  E. z A. y ( y  e.  z  <->  E. x  e.  a 
ph ) )
2 nfv 1466 . . . . 5  |-  F/ b  y  e.  z
3 nfcv 2228 . . . . . 6  |-  F/_ b
a
4 strcollnf.nf . . . . . 6  |-  F/ b
ph
53, 4nfrexxy 2415 . . . . 5  |-  F/ b E. x  e.  a 
ph
62, 5nfbi 1526 . . . 4  |-  F/ b ( y  e.  z  <->  E. x  e.  a  ph )
76nfal 1513 . . 3  |-  F/ b A. y ( y  e.  z  <->  E. x  e.  a  ph )
8 nfv 1466 . . 3  |-  F/ z A. y ( y  e.  b  <->  E. x  e.  a  ph )
9 elequ2 1648 . . . . 5  |-  ( z  =  b  ->  (
y  e.  z  <->  y  e.  b ) )
109bibi1d 231 . . . 4  |-  ( z  =  b  ->  (
( y  e.  z  <->  E. x  e.  a  ph )  <->  ( y  e.  b  <->  E. x  e.  a 
ph ) ) )
1110albidv 1752 . . 3  |-  ( z  =  b  ->  ( A. y ( y  e.  z  <->  E. x  e.  a 
ph )  <->  A. y
( y  e.  b  <->  E. x  e.  a  ph ) ) )
127, 8, 11cbvex 1686 . 2  |-  ( E. z A. y ( y  e.  z  <->  E. x  e.  a  ph )  <->  E. b A. y ( y  e.  b  <->  E. x  e.  a 
ph ) )
131, 12sylib 120 1  |-  ( A. x  e.  a  E. y ph  ->  E. b A. y ( y  e.  b  <->  E. x  e.  a 
ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   F/wnf 1394   E.wex 1426   A.wral 2359   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-strcoll 11523
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365
This theorem is referenced by: (None)
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