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Theorem sbequ8 1711
Description: Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)
Assertion
Ref Expression
sbequ8  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )

Proof of Theorem sbequ8
StepHypRef Expression
1 pm5.4 362 . . . 4  |-  ( ( x  =  y  -> 
( x  =  y  ->  ph ) )  <->  ( x  =  y  ->  ph )
)
21bicomi 202 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ( x  =  y  ->  ph )
) )
3 abai 793 . . . 4  |-  ( ( x  =  y  /\  ph )  <->  ( x  =  y  /\  ( x  =  y  ->  ph )
) )
43exbii 1639 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  y  /\  ( x  =  y  ->  ph )
) )
52, 4anbi12i 697 . 2  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  <->  ( ( x  =  y  ->  (
x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ( x  =  y  ->  ph )
) ) )
6 df-sb 1707 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
7 df-sb 1707 . 2  |-  ( [ y  /  x ]
( x  =  y  ->  ph )  <->  ( (
x  =  y  -> 
( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ( x  =  y  ->  ph )
) ) )
85, 6, 73bitr4i 277 1  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1591   [wsb 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-sb 1707
This theorem is referenced by: (None)
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