ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralxfrd Unicode version

Theorem ralxfrd 4445
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfrd.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
ralxfrd.2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
ralxfrd.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralxfrd  |-  ( 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 ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
2 ralxfrd.3 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32adantlr 474 . . . 4  |-  ( ( ( ph  /\  y  e.  C )  /\  x  =  A )  ->  ( ps 
<->  ch ) )
41, 3rspcdv 2837 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( A. x  e.  B  ps  ->  ch ) )
54ralrimdva 2550 . 2  |-  ( ph  ->  ( A. x  e.  B  ps  ->  A. y  e.  C  ch )
)
6 ralxfrd.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
7 r19.29 2607 . . . . 5  |-  ( ( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  E. y  e.  C  ( ch  /\  x  =  A ) )
82biimprd 157 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
98expimpd 361 . . . . . . . 8  |-  ( ph  ->  ( ( x  =  A  /\  ch )  ->  ps ) )
109ancomsd 267 . . . . . . 7  |-  ( ph  ->  ( ( ch  /\  x  =  A )  ->  ps ) )
1110ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
( ch  /\  x  =  A )  ->  ps ) )
1211rexlimdva 2587 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( E. y  e.  C  ( ch  /\  x  =  A )  ->  ps ) )
137, 12syl5 32 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  ps ) )
146, 13mpan2d 426 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( A. y  e.  C  ch  ->  ps ) )
1514ralrimdva 2550 . 2  |-  ( ph  ->  ( A. y  e.  C  ch  ->  A. x  e.  B  ps )
)
165, 15impbid 128 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 1348    e. wcel 2141   A.wral 2448   E.wrex 2449
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732
This theorem is referenced by:  ralxfr2d  4447  ralxfr  4449
  Copyright terms: Public domain W3C validator