Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ralxfrd2 Structured version   Unicode version

Theorem ralxfrd2 32137
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. Variant of ralxfrd 4647. (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 1195 . . . 4  |-  ( ( ( ph  /\  y  e.  C )  /\  x  =  A )  ->  ( ps 
<->  ch ) )
41, 3rspcdv 3197 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( A. x  e.  B  ps  ->  ch ) )
54ralrimdva 2859 . 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 2976 . . . . 5  |-  ( ( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  E. y  e.  C  ( ch  /\  x  =  A ) )
8 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  ph )
9 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  y  e.  C )
10 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  x  =  A )
118, 9, 10, 2syl3anc 1227 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  C
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1211exbiri 622 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
x  =  A  -> 
( ch  ->  ps ) ) )
1312com23 78 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  ( ch  ->  ( x  =  A  ->  ps )
) )
1413impd 431 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
( ch  /\  x  =  A )  ->  ps ) )
1514rexlimdva 2933 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( E. y  e.  C  ( ch  /\  x  =  A )  ->  ps ) )
167, 15syl5 32 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  ps ) )
176, 16mpan2d 674 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( A. y  e.  C  ch  ->  ps ) )
1817ralrimdva 2859 . 2  |-  ( ph  ->  ( A. y  e.  C  ch  ->  A. x  e.  B  ps )
)
195, 18impbid 191 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-v 3095
This theorem is referenced by:  rexxfrd2  32138
  Copyright terms: Public domain W3C validator