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Theorem ralxfrALT 4553
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4548. (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 2880 . . . . 5  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
41, 3syl 15 . . . 4  |-  ( y  e.  C  ->  ( A. x  e.  B  ph 
->  ps ) )
54com12 27 . . 3  |-  ( A. x  e.  B  ph  ->  ( y  e.  C  ->  ps ) )
65ralrimiv 2625 . 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 2593 . . . . 5  |-  F/ y A. y  e.  C  ps
9 nfv 1605 . . . . 5  |-  F/ y
ph
10 rsp 2603 . . . . . 6  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ps ) )
112biimprcd 216 . . . . . 6  |-  ( ps 
->  ( x  =  A  ->  ph ) )
1210, 11syl6 29 . . . . 5  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ( x  =  A  ->  ph ) ) )
138, 9, 12rexlimd 2664 . . . 4  |-  ( A. y  e.  C  ps  ->  ( E. y  e.  C  x  =  A  ->  ph ) )
147, 13syl5 28 . . 3  |-  ( A. y  e.  C  ps  ->  ( x  e.  B  ->  ph ) )
1514ralrimiv 2625 . 2  |-  ( A. y  e.  C  ps  ->  A. x  e.  B  ph )
166, 15impbii 180 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544
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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