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Theorem ralxfrALT 4771
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4766. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
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
ralxfrALT  |-  ( 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 ralxfrALT
StepHypRef Expression
1 ralxfr.1 . . . . 5  |-  ( y  e.  C  ->  A  e.  B )
2 ralxfr.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32rspcv 3054 . . . . 5  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
41, 3syl 16 . . . 4  |-  ( y  e.  C  ->  ( A. x  e.  B  ph 
->  ps ) )
54com12 30 . . 3  |-  ( A. x  e.  B  ph  ->  ( y  e.  C  ->  ps ) )
65ralrimiv 2794 . 2  |-  ( A. x  e.  B  ph  ->  A. y  e.  C  ps )
7 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
8 nfra1 2762 . . . . 5  |-  F/ y A. y  e.  C  ps
9 nfv 1630 . . . . 5  |-  F/ y
ph
10 rsp 2772 . . . . . 6  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ps ) )
112biimprcd 218 . . . . . 6  |-  ( ps 
->  ( x  =  A  ->  ph ) )
1210, 11syl6 32 . . . . 5  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ( x  =  A  ->  ph ) ) )
138, 9, 12rexlimd 2833 . . . 4  |-  ( A. y  e.  C  ps  ->  ( E. y  e.  C  x  =  A  ->  ph ) )
147, 13syl5 31 . . 3  |-  ( A. y  e.  C  ps  ->  ( x  e.  B  ->  ph ) )
1514ralrimiv 2794 . 2  |-  ( A. y  e.  C  ps  ->  A. x  e.  B  ph )
166, 15impbii 182 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-v 2964
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