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Theorem ralxfrd2 39008
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. Variant of ralxfrd 4614. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
Hypotheses
Ref Expression
ralxfrd2.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
ralxfrd2.2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
ralxfrd2.3  |-  ( (
ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps  <->  ch ) )
Assertion
Ref Expression
ralxfrd2  |-  ( 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)

Proof of Theorem ralxfrd2
StepHypRef Expression
1 ralxfrd2.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
2 ralxfrd2.3 . . . . 5  |-  ( (
ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps  <->  ch ) )
323expa 1208 . . . 4  |-  ( ( ( ph  /\  y  e.  C )  /\  x  =  A )  ->  ( ps 
<->  ch ) )
41, 3rspcdv 3153 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( A. x  e.  B  ps  ->  ch ) )
54ralrimdva 2806 . 2  |-  ( ph  ->  ( A. x  e.  B  ps  ->  A. y  e.  C  ch )
)
6 ralxfrd2.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
7 r19.29 2925 . . . . 5  |-  ( ( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  E. y  e.  C  ( ch  /\  x  =  A ) )
8 simplll 768 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  ph )
9 simplr 762 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  y  e.  C )
10 simpr 463 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  x  =  A )
118, 9, 10, 2syl3anc 1268 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1211exbiri 628 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
x  =  A  -> 
( ch  ->  ps ) ) )
1312com23 81 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  ( ch  ->  ( x  =  A  ->  ps )
) )
1413impd 433 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
( ch  /\  x  =  A )  ->  ps ) )
1514rexlimdva 2879 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( E. y  e.  C  ( ch  /\  x  =  A )  ->  ps ) )
167, 15syl5 33 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  ps ) )
176, 16mpan2d 680 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( A. y  e.  C  ch  ->  ps ) )
1817ralrimdva 2806 . 2  |-  ( ph  ->  ( A. y  e.  C  ch  ->  A. x  e.  B  ps )
)
195, 18impbid 194 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-v 3047
This theorem is referenced by:  rexxfrd2  39009
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