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Theorem nrexdv 2455
Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
Hypothesis
Ref Expression
nrexdv.1  |-  ( (
ph  /\  x  e.  A )  ->  -.  ps )
Assertion
Ref Expression
nrexdv  |-  ( ph  ->  -.  E. x  e.  A  ps )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem nrexdv
StepHypRef Expression
1 nrexdv.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  -.  ps )
21ralrimiva 2435 . 2  |-  ( ph  ->  A. x  e.  A  -.  ps )
3 ralnex 2359 . 2  |-  ( A. x  e.  A  -.  ps 
<->  -.  E. x  e.  A  ps )
42, 3sylib 120 1  |-  ( ph  ->  -.  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2349   E.wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  ltpopr  6847  cauappcvgprlemladdru  6908  cauappcvgprlemladdrl  6909  caucvgprlemladdrl  6930  caucvgprprlemaddq  6960  dvdsle  10389
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