Theorem sb5 1841

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

Step	Hyp	Ref	Expression
1		sb6 1840	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb56 1839	. 2 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
3	1, 2	bitr4i 186	1 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1312   E.wex 1451   [wsb 1718

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497

This theorem depends on definitions:  df-bi 116  df-sb 1719

This theorem is referenced by:  sbnv  1842  sborv  1844  sbi2v  1846  nfsbxy  1893  nfsbxyt  1894  2sb5  1934  dfsb7  1942  sb7f  1943  sbexyz  1954  sbc5  2901