Theorem sb5rf 1866

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 1863	. . 3 |- ( [ x / y ] [ y / x ] ph <-> ph )
3		sb1 1780	. . 3 |- ( [ x / y ] [ y / x ] ph -> E. y ( y = x /\ [ y / x ] ph ) )
4	2, 3	sylbir 135	. 2 |- ( ph -> E. y ( y = x /\ [ y / x ] ph ) )
5		stdpc7 1784	. . . 4 |- ( y = x -> ( [ y / x ] ph -> ph ) )
6	5	imp 124	. . 3 |- ( ( y = x /\ [ y / x ] ph ) -> ph )
7	1, 6	exlimih 1607	. 2 |- ( E. y ( y = x /\ [ y / x ] ph ) -> ph )
8	4, 7	impbii 126	1 |- ( ph <-> E. y ( y = x /\ [ y / x ] 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-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

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

This theorem is referenced by:  2sb5rf  2008  sbelx  2016