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Theorem alxfr 4547
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 2868 . . . . . 6  |-  ( A  e.  B  ->  ( A. x ph  ->  ps ) )
32com12 27 . . . . 5  |-  ( A. x ph  ->  ( A  e.  B  ->  ps )
)
43alimdv 1607 . . . 4  |-  ( A. x ph  ->  ( A. y  A  e.  B  ->  A. y ps )
)
54com12 27 . . 3  |-  ( A. y  A  e.  B  ->  ( A. x ph  ->  A. y ps )
)
65adantr 451 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  ->  A. y ps ) )
7 nfa1 1756 . . . . . 6  |-  F/ y A. y ps
8 nfv 1605 . . . . . 6  |-  F/ y
ph
9 sp 1716 . . . . . . 7  |-  ( A. y ps  ->  ps )
109, 1syl5ibrcom 213 . . . . . 6  |-  ( A. y ps  ->  ( x  =  A  ->  ph )
)
117, 8, 10exlimd 1803 . . . . 5  |-  ( A. y ps  ->  ( E. y  x  =  A  ->  ph ) )
1211alimdv 1607 . . . 4  |-  ( A. y ps  ->  ( A. x E. y  x  =  A  ->  A. x ph ) )
1312com12 27 . . 3  |-  ( A. x E. y  x  =  A  ->  ( A. y ps  ->  A. x ph ) )
1413adantl 452 . 2  |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. y ps  ->  A. x ph ) )
156, 14impbid 183 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    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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