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Theorem alxfr 4350
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 2745 . . . . . 6  |-  ( A  e.  B  ->  ( A. x ph  ->  ps ) )
32com12 30 . . . . 5  |-  ( A. x ph  ->  ( A  e.  B  ->  ps )
)
43alimdv 1833 . . . 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 272 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  ->  A. y ps ) )
7 nfa1 1504 . . . . . 6  |-  F/ y A. y ps
8 nfv 1491 . . . . . 6  |-  F/ y
ph
9 sp 1471 . . . . . . 7  |-  ( A. y ps  ->  ps )
109, 1syl5ibrcom 156 . . . . . 6  |-  ( A. y ps  ->  ( x  =  A  ->  ph )
)
117, 8, 10exlimd 1559 . . . . 5  |-  ( A. y ps  ->  ( E. y  x  =  A  ->  ph ) )
1211alimdv 1833 . . . 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 273 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. y ps  ->  A. x ph ) )
156, 14impbid 128 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 103    <-> wb 104   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by: (None)
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