Theorem sbh 1699

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)

Hypothesis

Ref	Expression
sbh.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
sbh	|- ( [ y / x ] ph <-> ph )

Proof of Theorem sbh

Step	Hyp	Ref	Expression
1		sb1 1689	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1615	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ( E. x x = y /\ ph ) )
4	1, 3	sylib 120	. . 3 |- ( [ y / x ] ph -> ( E. x x = y /\ ph ) )
5	4	simprd 112	. 2 |- ( [ y / x ] ph -> ph )
6		stdpc4 1698	. . 3 |- ( A. x ph -> [ y / x ] ph )
7	2, 6	syl 14	. 2 |- ( ph -> [ y / x ] ph )
8	5, 7	impbii 124	1 |- ( [ y / x ] ph <-> ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sbf  1700  sb6x  1702  nfs1f  1703  hbs1f  1704  sbid2h  1770  sblimv  1815  sbrim  1871  sbrbif  1877  elsb3  1893  elsb4  1894