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Theorem ralxfrALT 4263
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4258. (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 2711 . . . . 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 2441 . 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 2405 . . . . 5  |-  F/ y A. y  e.  C  ps
9 nfv 1464 . . . . 5  |-  F/ y
ph
10 rsp 2419 . . . . . 6  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ps ) )
112biimprcd 158 . . . . . 6  |-  ( ps 
->  ( x  =  A  ->  ph ) )
1210, 11syl6 33 . . . . 5  |-  ( A. y  e.  C  ps  ->  ( y  e.  C  ->  ( x  =  A  ->  ph ) ) )
138, 9, 12rexlimd 2482 . . . 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 2441 . 2  |-  ( A. y  e.  C  ps  ->  A. x  e.  B  ph )
166, 15impbii 124 1  |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617
This theorem is referenced by: (None)
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