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Theorem ralxfr 4552
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 452 . . 3  |-  ( (  T.  /\  y  e.  C )  ->  A  e.  B )
3 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
43adantl 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 ralxfr.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65adantl 452 . . 3  |-  ( (  T.  /\  x  =  A )  ->  ( ph 
<->  ps ) )
72, 4, 6ralxfrd 4548 . 2  |-  (  T. 
->  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
)
87trud 1314 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    T. wtru 1307    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544
This theorem is referenced by:  rexxfr  4554  infm3  9713
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-ral 2548  df-rex 2549  df-v 2790
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