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Theorem sb5 1809
Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
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 1808 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 sb56 1807 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
31, 2bitr4i 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-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbnv  1810  sborv  1812  sbi2v  1814  nfsbxy  1860  nfsbxyt  1861  2sb5  1901  dfsb7  1909  sb7f  1910  sbexyz  1921  sbc5  2839
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