Theorem sb5 1859

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 1858	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb56 1857	. 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 1329   E.wex 1468   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sbnv  1860  sborv  1862  sbi2v  1864  nfsbxy  1913  nfsbxyt  1914  2sb5  1956  dfsb7  1964  sb7f  1965  sbexyz  1976  sbc5  2927