MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reubidv Unicode version

Theorem reubidv 2724
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
Hypothesis
Ref Expression
reubidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
reubidv  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem reubidv
StepHypRef Expression
1 reubidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21adantr 451 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32reubidva 2723 1  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   E!wreu 2545
This theorem is referenced by:  reueqd  2746  sbcreug  3067  reusv6OLD  4545  reusv7OLD  4546  riotabidv  6306  oawordeu  6553  xpf1o  7023  dfac2  7757  creur  9740  creui  9741  divalg  12602  divalg2  12604  spwpr4  14340  dfod2  14877  riesz4  22644  cnlnadjeu  22658  reubidvag  24935  isig2a2  26066  isibcg  26191  isfrgra  28171  frgra1v  28176  frgra3v  28180  3vfriswmgra  28183  hdmap1eulem  32014  hdmap1eulemOLDN  32015  hdmap14lem6  32066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-eu 2147  df-reu 2550
  Copyright terms: Public domain W3C validator