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Theorem strcollnfALT 10498
Description: Alternate proof of strcollnf 10497, not using strcollnft 10496. (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 10495 . 2  |-  ( A. x  e.  a  E. y ph  ->  E. z A. y ( y  e.  z  <->  E. x  e.  a 
ph ) )
2 nfv 1437 . . . . 5  |-  F/ b  y  e.  z
3 nfcv 2194 . . . . . 6  |-  F/_ b
a
4 strcollnf.nf . . . . . 6  |-  F/ b
ph
53, 4nfrexxy 2378 . . . . 5  |-  F/ b E. x  e.  a 
ph
62, 5nfbi 1497 . . . 4  |-  F/ b ( y  e.  z  <->  E. x  e.  a  ph )
76nfal 1484 . . 3  |-  F/ b A. y ( y  e.  z  <->  E. x  e.  a  ph )
8 nfv 1437 . . 3  |-  F/ z A. y ( y  e.  b  <->  E. x  e.  a  ph )
9 elequ2 1617 . . . . 5  |-  ( z  =  b  ->  (
y  e.  z  <->  y  e.  b ) )
109bibi1d 226 . . . 4  |-  ( z  =  b  ->  (
( y  e.  z  <->  E. x  e.  a  ph )  <->  ( y  e.  b  <->  E. x  e.  a 
ph ) ) )
1110albidv 1721 . . 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 1655 . 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 131 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 102   A.wal 1257   F/wnf 1365   E.wex 1397   A.wral 2323   E.wrex 2324
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-strcoll 10494
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329
This theorem is referenced by: (None)
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