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Theorem alxfr 4240
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 18-Feb-2007.)
Hypothesis
Ref Expression
alxfr.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
alxfr  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  <->  A. y ps ) )
Distinct variable groups:    x, A    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)    A( y)    B( x, y)

Proof of Theorem alxfr
StepHypRef Expression
1 alxfr.1 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21spcgv 2694 . . . . . 6  |-  ( A  e.  B  ->  ( A. x ph  ->  ps ) )
32com12 30 . . . . 5  |-  ( A. x ph  ->  ( A  e.  B  ->  ps )
)
43alimdv 1802 . . . 4  |-  ( A. x ph  ->  ( A. y  A  e.  B  ->  A. y ps )
)
54com12 30 . . 3  |-  ( A. y  A  e.  B  ->  ( A. x ph  ->  A. y ps )
)
65adantr 270 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  ->  A. y ps ) )
7 nfa1 1475 . . . . . 6  |-  F/ y A. y ps
8 nfv 1462 . . . . . 6  |-  F/ y
ph
9 sp 1442 . . . . . . 7  |-  ( A. y ps  ->  ps )
109, 1syl5ibrcom 155 . . . . . 6  |-  ( A. y ps  ->  ( x  =  A  ->  ph )
)
117, 8, 10exlimd 1529 . . . . 5  |-  ( A. y ps  ->  ( E. y  x  =  A  ->  ph ) )
1211alimdv 1802 . . . 4  |-  ( A. y ps  ->  ( A. x E. y  x  =  A  ->  A. x ph ) )
1312com12 30 . . 3  |-  ( A. x E. y  x  =  A  ->  ( A. y ps  ->  A. x ph ) )
1413adantl 271 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. y ps  ->  A. x ph ) )
156, 14impbid 127 1  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by: (None)
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