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Theorem sb6a 2279
Description: Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.)
Assertion
Ref Expression
sb6a  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  [ x  /  y ] ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb6a
StepHypRef Expression
1 sbco 2243 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
2 sb6 2260 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph ) )
31, 2bitr3i 255 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  [ x  /  y ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1444   [wsb 1799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-12 1935  ax-13 2093
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800
This theorem is referenced by: (None)
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