Theorem sb6 1815

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 1814	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
2	1	anbi2i 446	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
3		df-sb 1694	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4		ax-4 1446	. . 3 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
5	4	pm4.71ri 385	. 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 1288   E.wex 1427   [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-sb 1694

This theorem is referenced by:  sb5  1816  sbnv  1817  sbanv  1818  sbi1v  1820  sbi2v  1821  hbs1  1863  2sb6  1909  sbcom2v  1910  sb6a  1913  sb7af  1918  sbalyz  1924  sbal1yz  1926  exsb  1933  sbal2  1947