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Theorem ralxfr 4245
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfr.1  |-  ( y  e.  C  ->  A  e.  B )
ralxfr.2  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
ralxfr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralxfr  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Distinct variable groups:    ps, x    ph, y    x, A    x, y, B   
x, C
Allowed substitution hints:    ph( x)    ps( y)    A( y)    C( y)

Proof of Theorem ralxfr
StepHypRef Expression
1 ralxfr.1 . . . 4  |-  ( y  e.  C  ->  A  e.  B )
21adantl 271 . . 3  |-  ( ( T.  /\  y  e.  C )  ->  A  e.  B )
3 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
43adantl 271 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 ralxfr.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65adantl 271 . . 3  |-  ( ( T.  /\  x  =  A )  ->  ( ph 
<->  ps ) )
72, 4, 6ralxfrd 4241 . 2  |-  ( T. 
->  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
)
87trud 1294 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   T. wtru 1286    e. wcel 1434   A.wral 2353   E.wrex 2354
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-ral 2358  df-rex 2359  df-v 2613
This theorem is referenced by: (None)
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