Theorem sbh 1790

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 1780	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1699	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ( E. x x = y /\ ph ) )
4	1, 3	sylib 122	. . 3 |- ( [ y / x ] ph -> ( E. x x = y /\ ph ) )
5	4	simprd 114	. 2 |- ( [ y / x ] ph -> ph )
6		stdpc4 1789	. . 3 |- ( A. x ph -> [ y / x ] ph )
7	2, 6	syl 14	. 2 |- ( ph -> [ y / x ] ph )
8	5, 7	impbii 126	1 |- ( [ y / x ] ph <-> ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506   [wsb 1776

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  sbf  1791  sb6x  1793  nfs1f  1794  hbs1f  1795  sbid2h  1863  sblimv  1909  sbrim  1975  sbrbif  1981  elsb1  2174  elsb2  2175