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Theorem sb5f 1726
Description: Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)
Hypothesis
Ref Expression
equs45f.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb5f  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )

Proof of Theorem sb5f
StepHypRef Expression
1 equs45f.1 . . 3  |-  ( ph  ->  A. y ph )
21sb6f 1725 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
31equs45f 1724 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
42, 3bitr4i 185 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-11 1438  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbcof2  1732
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