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Theorem nnssi3 26208
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 3346 . . 3  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3346 . . 3  |-  ( B  e.  NN  ->  B  e.  D )
41sseli 3346 . . 3  |-  ( C  e.  NN  ->  C  e.  D )
52, 3, 43anim123i 1140 . 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 981 . 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 644 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726    C_ wss 3322   NNcn 10002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336
  Copyright terms: Public domain W3C validator