Theorem sb6rf 1825

Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

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

Assertion

Ref	Expression
sb6rf	|- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )

Proof of Theorem sb6rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . 3 |- ( ph -> A. y ph )
2		sbequ1 1741	. . . . 5 |- ( x = y -> ( ph -> [ y / x ] ph ) )
3	2	equcoms 1684	. . . 4 |- ( y = x -> ( ph -> [ y / x ] ph ) )
4	3	com12 30	. . 3 |- ( ph -> ( y = x -> [ y / x ] ph ) )
5	1, 4	alrimih 1445	. 2 |- ( ph -> A. y ( y = x -> [ y / x ] ph ) )
6		sb2 1740	. . 3 |- ( A. y ( y = x -> [ y / x ] ph ) -> [ x / y ] [ y / x ] ph )
7	1	sbid2h 1821	. . 3 |- ( [ x / y ] [ y / x ] ph <-> ph )
8	6, 7	sylib 121	. 2 |- ( A. y ( y = x -> [ y / x ] ph ) -> ph )
9	5, 8	impbii 125	1 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   [wsb 1735

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  2sb6rf  1965  eu1  2024