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Theorem sb5 1993
Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
sb5  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb5
StepHypRef Expression
1 sb6 1992 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 sb56 1991 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
31, 2bitr4i 245 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537   [wsb 1882
This theorem is referenced by:  2sb5  2072  dfsb7  2079  sb7f  2080  sbelx  2084  sbc2or  2929  sbc5  2945
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883
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