Theorem sbh 1800

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 1790	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1709	. . . 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 1799	. . 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 1371   E.wex 1516   [wsb 1786

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by:  sbf  1801  sb6x  1803  nfs1f  1804  hbs1f  1805  sbid2h  1873  sblimv  1919  sbrim  1985  sbrbif  1991  elsb1  2185  elsb2  2186