Theorem sb6 1898

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 1897	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
2	1	anbi2i 457	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
3		df-sb 1774	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4		ax-4 1521	. . 3 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
5	4	pm4.71ri 392	. 2 |- ( A. x ( x = y -> ph ) <-> ( ( x = y -> ph ) /\ A. x ( x = y -> ph ) ) )
6	2, 3, 5	3bitr4i 212	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1503   [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  sb5  1899  sbnv  1900  sbanv  1901  sbi1v  1903  sbi2v  1904  hbs1  1954  2sb6  2000  sbcom2v  2001  sb6a  2004  sb7af  2009  sbalyz  2015  sbal1yz  2017  exsb  2024  sbal2  2036  cbvabw  2316  nfabdw  2355  csbcow  3091