Theorem sbh 1659

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 1649	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1575	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ( E. x x = y /\ ph ) )
4	1, 3	sylib 127	. . 3 |- ( [ y / x ] ph -> ( E. x x = y /\ ph ) )
5	4	simprd 107	. 2 |- ( [ y / x ] ph -> ph )
6		stdpc4 1658	. . 3 |- ( A. x ph -> [ y / x ] ph )
7	2, 6	syl 14	. 2 |- ( ph -> [ y / x ] ph )
8	5, 7	impbii 117	1 |- ( [ y / x ] ph <-> ph )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbf  1660  sb6x  1662  nfs1f  1663  hbs1f  1664  sbid2h  1729  sblimv  1774  sbrim  1830  sbrbif  1836  elsb3  1852  elsb4  1853