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Theorem ralxfr 4733
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 453 . . 3  |-  ( (  T.  /\  y  e.  C )  ->  A  e.  B )
3 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
43adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 ralxfr.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65adantl 453 . . 3  |-  ( (  T.  /\  x  =  A )  ->  ( ph 
<->  ps ) )
72, 4, 6ralxfrd 4729 . 2  |-  (  T. 
->  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
)
87trud 1332 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    T. wtru 1325    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698
This theorem is referenced by:  rexxfr  4735  infm3  9959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950
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