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Theorem ralxfr2d 4385
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.)
Hypotheses
Ref Expression
ralxfr2d.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
ralxfr2d.2  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
ralxfr2d.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralxfr2d  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Distinct variable groups:    x, A    x, y, B    x, C    ch, x    ph, x, y    ps, y
Allowed substitution hints:    ps( x)    ch( y)    A( y)    C( y)    V( x, y)

Proof of Theorem ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
2 elisset 2700 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
31, 2syl 14 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  E. x  x  =  A )
4 ralxfr2d.2 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
54biimprd 157 . . . . . . 7  |-  ( ph  ->  ( E. y  e.  C  x  =  A  ->  x  e.  B
) )
6 r19.23v 2541 . . . . . . 7  |-  ( A. y  e.  C  (
x  =  A  ->  x  e.  B )  <->  ( E. y  e.  C  x  =  A  ->  x  e.  B ) )
75, 6sylibr 133 . . . . . 6  |-  ( ph  ->  A. y  e.  C  ( x  =  A  ->  x  e.  B ) )
87r19.21bi 2520 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  x  e.  B )
)
9 eleq1 2202 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
108, 9mpbidi 150 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  A  e.  B )
)
1110exlimdv 1791 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  x  =  A  ->  A  e.  B
) )
123, 11mpd 13 . 2  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
134biimpa 294 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
14 ralxfr2d.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1512, 13, 14ralxfrd 4383 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688
This theorem is referenced by:  ralrn  5558  ralima  5657  cnrest2  12415  cnptoprest2  12419
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