Theorem sb5 1942

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 1941	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb56 1940	. 2 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
3	1, 2	bitr4i 187	1 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   E.wex 1545   [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-sb 1816

This theorem is referenced by:  sbnv  1943  sborv  1945  sbi2v  1947  nfsbxy  2002  nfsbxyt  2003  2sb5  2043  dfsb7  2051  sb7f  2052  sbexyz  2063  sbc5  3075