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Theorem sb6 1814
Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb6  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb6
StepHypRef Expression
1 sb56 1813 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
21anbi2i 445 . 2  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  <->  ( ( x  =  y  ->  ph )  /\  A. x ( x  =  y  ->  ph )
) )
3 df-sb 1693 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
4 ax-4 1445 . . 3  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
54pm4.71ri 384 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  ( (
x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ph ) ) )
62, 3, 53bitr4i 210 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426   [wsb 1692
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  sb5  1815  sbnv  1816  sbanv  1817  sbi1v  1819  sbi2v  1820  hbs1  1862  2sb6  1908  sbcom2v  1909  sb6a  1912  sb7af  1917  sbalyz  1923  sbal1yz  1925  exsb  1932  sbal2  1946
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