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Theorem nnssi3 30701
Description: Convert a theorem for real/complex numbers into one for positive integers. (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 3440 . . 3  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3440 . . 3  |-  ( B  e.  NN  ->  B  e.  D )
41sseli 3440 . . 3  |-  ( C  e.  NN  ->  C  e.  D )
52, 3, 43anim123i 1184 . 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 1022 . 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 661 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    e. wcel 1844    C_ wss 3416   NNcn 10578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-in 3423  df-ss 3430
This theorem is referenced by: (None)
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