Theorem sb6 1763

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 1762	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
2	1	anbi2i 430	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
3		df-sb 1643	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
4		ax-4 1397	. . 3 |- (A.x(x = y -> ph) -> (x = y -> ph))
5	4	pm4.71ri 372	. 2 |- (A.x(x = y -> ph) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
6	2, 3, 5	3bitr4i 201	1 |- ([y / x]ph <-> A.x(x = y -> ph))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sb5  1764  sbnv  1765  sbanv  1766  sbi1v  1768  sbi2v  1769  hbs1  1811  2sb6  1857  sbcom2v  1858  sb6a  1861  sb7af  1866  sbalyz  1872  sbal1yz  1874  exsb  1881  sbal2  1895