Theorem sb6 1807

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

Step	Hyp	Ref	Expression
1		sb56 1806	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
2	1	anbi2i 444	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
3		df-sb 1686	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4		ax-4 1440	. . 3 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
5	4	pm4.71ri 384	. 2 |- ( A. x ( x = y -> ph ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
6	2, 3, 5	3bitr4i 210	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sb5  1808  sbnv  1809  sbanv  1810  sbi1v  1812  sbi2v  1813  hbs1  1855  2sb6  1901  sbcom2v  1902  sb6a  1905  sb7af  1910  sbalyz  1916  sbal1yz  1918  exsb  1925  sbal2  1939