Theorem sbh 1825

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 1815	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sbh.1	. . . . 5 |- ( ph -> A. x ph )
3	2	19.41h 1733	. . . 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 1824	. . 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 1396   E.wex 1541   [wsb 1811

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-sb 1812

This theorem is referenced by:  sbf  1826  sb6x  1828  nfs1f  1829  hbs1f  1830  sbid2h  1898  sblimv  1944  sbrim  2010  sbrbif  2016  elsb1  2210  elsb2  2211