Theorem 2sb6rf 2009

Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)

Hypotheses

Ref	Expression
2sb5rf.1	|- ( ph -> A. z ph )
2sb5rf.2	|- ( ph -> A. w ph )

Assertion

Ref	Expression
2sb6rf	|- ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )

Distinct variable groups:   x, y   x, w   y, z   z, w

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem 2sb6rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 |- ( ph -> A. z ph )
2	1	sb6rf 1867	. 2 |- ( ph <-> A. z ( z = x -> [ z / x ] ph ) )
3		19.21v 1887	. . . 4 |- ( A. w ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) <-> ( z = x -> A. w ( w = y -> [ w / y ] [ z / x ] ph ) ) )
4		sbcom2 2006	. . . . . . 7 |- ( [ z / x ] [ w / y ] ph <-> [ w / y ] [ z / x ] ph )
5	4	imbi2i 226	. . . . . 6 |- ( ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> ( ( z = x /\ w = y ) -> [ w / y ] [ z / x ] ph ) )
6		impexp 263	. . . . . 6 |- ( ( ( z = x /\ w = y ) -> [ w / y ] [ z / x ] ph ) <-> ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) )
7	5, 6	bitri 184	. . . . 5 |- ( ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) )
8	7	albii 1484	. . . 4 |- ( A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> A. w ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) )
9		2sb5rf.2	. . . . . . 7 |- ( ph -> A. w ph )
10	9	hbsbv 1960	. . . . . 6 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
11	10	sb6rf 1867	. . . . 5 |- ( [ z / x ] ph <-> A. w ( w = y -> [ w / y ] [ z / x ] ph ) )
12	11	imbi2i 226	. . . 4 |- ( ( z = x -> [ z / x ] ph ) <-> ( z = x -> A. w ( w = y -> [ w / y ] [ z / x ] ph ) ) )
13	3, 8, 12	3bitr4ri 213	. . 3 |- ( ( z = x -> [ z / x ] ph ) <-> A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )
14	13	albii 1484	. 2 |- ( A. z ( z = x -> [ z / x ] ph ) <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )
15	2, 14	bitri 184	1 |- ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   [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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by: (None)