Theorem sbh 1756

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 1746	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1665	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ( E. x x = y /\ ph ) )
4	1, 3	sylib 121	. . 3 |- ( [ y / x ] ph -> ( E. x x = y /\ ph ) )
5	4	simprd 113	. 2 |- ( [ y / x ] ph -> ph )
6		stdpc4 1755	. . 3 |- ( A. x ph -> [ y / x ] ph )
7	2, 6	syl 14	. 2 |- ( ph -> [ y / x ] ph )
8	5, 7	impbii 125	1 |- ( [ y / x ] ph <-> ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   E.wex 1472   [wsb 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1743

This theorem is referenced by:  sbf  1757  sb6x  1759  nfs1f  1760  hbs1f  1761  sbid2h  1829  sblimv  1874  sbrim  1936  sbrbif  1942  elsb3  2135  elsb4  2136