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Theorem nnssi3 24895
Description: Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
nnssi3.1  |-  NN  C_  D
nnssi3.2  |-  ( C  e.  NN  ->  ph )
nnssi3.3  |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D
)  /\  ph )  ->  ps )
Assertion
Ref Expression
nnssi3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )

Proof of Theorem nnssi3
StepHypRef Expression
1 nnssi3.1 . . . 4  |-  NN  C_  D
21sseli 3176 . . 3  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3176 . . 3  |-  ( B  e.  NN  ->  B  e.  D )
41sseli 3176 . . 3  |-  ( C  e.  NN  ->  C  e.  D )
52, 3, 43anim123i 1137 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  C  e.  D )
)
6 nnssi3.2 . . 3  |-  ( C  e.  NN  ->  ph )
763ad2ant3 978 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ph )
8 nnssi3.3 . 2  |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D
)  /\  ph )  ->  ps )
95, 7, 8syl2anc 642 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   NNcn 9746
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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