Theorem sb5 1911

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 1910	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb56 1909	. 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 1371   E.wex 1515   [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  sbnv  1912  sborv  1914  sbi2v  1916  nfsbxy  1970  nfsbxyt  1971  2sb5  2011  dfsb7  2019  sb7f  2020  sbexyz  2031  sbc5  3022