Theorem 2sb6rf 1908

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 1775	. 2 |- ( ph <-> A. z ( z = x -> [ z / x ] ph ) )
3		19.21v 1795	. . . 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 1905	. . . . . . 7 |- ( [ z / x ] [ w / y ] ph <-> [ w / y ] [ z / x ] ph )
5	4	imbi2i 224	. . . . . 6 |- ( ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> ( ( z = x /\ w = y ) -> [ w / y ] [ z / x ] ph ) )
6		impexp 259	. . . . . 6 |- ( ( ( z = x /\ w = y ) -> [ w / y ] [ z / x ] ph ) <-> ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) )
7	5, 6	bitri 182	. . . . 5 |- ( ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> ( z = x -> ( w = y -> [ w / y ] [ z / x ] ph ) ) )
8	7	albii 1400	. . . 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 1859	. . . . . 6 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
11	10	sb6rf 1775	. . . . 5 |- ( [ z / x ] ph <-> A. w ( w = y -> [ w / y ] [ z / x ] ph ) )
12	11	imbi2i 224	. . . 4 |- ( ( z = x -> [ z / x ] ph ) <-> ( z = x -> A. w ( w = y -> [ w / y ] [ z / x ] ph ) ) )
13	3, 8, 12	3bitr4ri 211	. . 3 |- ( ( z = x -> [ z / x ] ph ) <-> A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )
14	13	albii 1400	. 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 182	1 |- ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by: (None)