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Theorem dfsb3 2223
Description: An alternate definition of proper substitution df-sb 1806 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
Assertion
Ref Expression
dfsb3  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem dfsb3
StepHypRef Expression
1 df-or 377 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  <->  ( -.  (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 dfsb2 2222 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
3 imnan 429 . . 3  |-  ( ( x  =  y  ->  -.  ph )  <->  -.  (
x  =  y  /\  ph ) )
43imbi1i 332 . 2  |-  ( ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph ) )  <-> 
( -.  ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
51, 2, 43bitr4i 285 1  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376   A.wal 1450   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbn  2240
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