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Theorem sbequ8ALT 2256
Description: Alternate proof of sbequ8 1810, shorter but requiring more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbequ8ALT  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )

Proof of Theorem sbequ8ALT
StepHypRef Expression
1 equsb1 2217 . . 3  |-  [ y  /  x ] x  =  y
21a1bi 344 . 2  |-  ( [ y  /  x ] ph 
<->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ph ) )
3 sbim 2244 . 2  |-  ( [ y  /  x ]
( x  =  y  ->  ph )  <->  ( [
y  /  x ]
x  =  y  ->  [ y  /  x ] ph ) )
42, 3bitr4i 260 1  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   [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: (None)
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