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

Theorem ceqsralv 3142
Description: Restricted quantifier version of ceqsalv 3141. (Contributed by NM, 21-Jun-2013.)
Hypothesis
Ref Expression
ceqsralv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsralv  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsralv
StepHypRef Expression
1 nfv 1683 . 2  |-  F/ x ps
2 ceqsralv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1601 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsralt 3137 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1314 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379   F/wnf 1599    e. wcel 1767   A.wral 2814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-v 3115
This theorem is referenced by:  eqreu  3295  sqrt2irr  13839  acsfn  14910  ovolgelb  21626
  Copyright terms: Public domain W3C validator