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Theorem dfsb3 1996
Description: An alternate definition of proper substitution df-sb 1630 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 359 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  <->  ( -.  (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 dfsb2 1995 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
3 imnan 411 . . 3  |-  ( ( x  =  y  ->  -.  ph )  <->  -.  (
x  =  y  /\  ph ) )
43imbi1i 315 . 2  |-  ( ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph ) )  <-> 
( -.  ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
51, 2, 43bitr4i 268 1  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527   [wsb 1629
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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