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Theorem dfsb3 2113
Description: An alternate definition of proper substitution df-sb 1660 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 361 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  <->  ( -.  (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 dfsb2 2112 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
3 imnan 413 . . 3  |-  ( ( x  =  y  ->  -.  ph )  <->  -.  (
x  =  y  /\  ph ) )
43imbi1i 317 . 2  |-  ( ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph ) )  <-> 
( -.  ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
51, 2, 43bitr4i 270 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 178    \/ wo 359    /\ wa 360   A.wal 1550   [wsb 1659
This theorem is referenced by:  sbn  2134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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