Theorem sb6 1859

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 1858	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
2	1	anbi2i 453	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
3		df-sb 1737	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4		ax-4 1488	. . 3 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
5	4	pm4.71ri 390	. 2 |- ( A. x ( x = y -> ph ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
6	2, 3, 5	3bitr4i 211	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   E.wex 1469   [wsb 1736

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-sb 1737

This theorem is referenced by:  sb5  1860  sbnv  1861  sbanv  1862  sbi1v  1864  sbi2v  1865  hbs1  1912  2sb6  1960  sbcom2v  1961  sb6a  1964  sb7af  1969  sbalyz  1975  sbal1yz  1977  exsb  1984  sbal2  1998