Theorem sb5rf 1732

Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sb5rf.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
sb5rf	|- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) )

Proof of Theorem sb5rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . 4 |- ( ph -> A. y ph )
2	1	sbid2h 1729	. . 3 |- ( [ x / y ] [ y / x ] ph <-> ph )
3		sb1 1649	. . 3 |- ( [ x / y ] [ y / x ] ph -> E. y ( y = x /\ [ y / x ] ph ) )
4	2, 3	sylbir 125	. 2 |- ( ph -> E. y ( y = x /\ [ y / x ] ph ) )
5		stdpc7 1653	. . . 4 |- ( y = x -> ( [ y / x ] ph -> ph ) )
6	5	imp 115	. . 3 |- ( ( y = x /\ [ y / x ] ph ) -> ph )
7	1, 6	exlimih 1484	. 2 |- ( E. y ( y = x /\ [ y / x ] ph ) -> ph )
8	4, 7	impbii 117	1 |- ( ph <-> E. y ( y = x /\ [ y / x ] 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-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

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

This theorem is referenced by:  2sb5rf  1865  sbelx  1873