Theorem sb6 1937

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 1936	. . 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 1812	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4		ax-4 1559	. . 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 1396   E.wex 1541   [wsb 1811

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-sb 1812

This theorem is referenced by:  sb5  1938  sbnv  1939  sbanv  1940  sbi1v  1942  sbi2v  1943  hbs1  1994  2sb6  2040  sbcom2v  2041  sb6a  2044  sb7af  2049  sbalyz  2055  sbal1yz  2057  exsb  2064  sbal2  2076  cbvabw  2359  nfabdw  2405  csbcow  3152