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Theorem ralxfrALT 4227
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4222. (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 2669 . . . . 5  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
41, 3syl 14 . . . 4  |-  ( y  e.  C  ->  ( A. x  e.  B  ph 
->  ps ) )
54com12 30 . . 3  |-  ( A. x  e.  B  ph  ->  ( y  e.  C  ->  ps ) )
65ralrimiv 2408 . 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 2372 . . . . 5  |-  F/ y A. y  e.  C  ps
9 nfv 1437 . . . . 5  |-  F/ y
ph
10 rsp 2386 . . . . . 6  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ps ) )
112biimprcd 153 . . . . . 6  |-  ( ps 
->  ( x  =  A  ->  ph ) )
1210, 11syl6 33 . . . . 5  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ( x  =  A  ->  ph ) ) )
138, 9, 12rexlimd 2447 . . . 4  |-  ( A. y  e.  C  ps  ->  ( E. y  e.  C  x  =  A  ->  ph ) )
147, 13syl5 32 . . 3  |-  ( A. y  e.  C  ps  ->  ( x  e.  B  ->  ph ) )
1514ralrimiv 2408 . 2  |-  ( A. y  e.  C  ps  ->  A. x  e.  B  ph )
166, 15impbii 121 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576
This theorem is referenced by: (None)
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