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Theorem ralxfrALT 4656
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4651. (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 2965 . . . . 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 2710 . 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 2678 . . . . 5  |-  F/ y A. y  e.  C  ps
9 nfv 1624 . . . . 5  |-  F/ y
ph
10 rsp 2688 . . . . . 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 2749 . . . 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 2710 . 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 1647    e. wcel 1715   A.wral 2628   E.wrex 2629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-v 2875
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