Theorem 2sb5rf 1906

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
2sb5rf	|- ( ph <-> E. z E. 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 2sb5rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 |- ( ph -> A. z ph )
2	1	sb5rf 1773	. 2 |- ( ph <-> E. z ( z = x /\ [ z / x ] ph ) )
3		19.42v 1827	. . . 4 |- ( E. w ( z = x /\ ( w = y /\ [ w / y ] [ z / x ] ph ) ) <-> ( z = x /\ E. w ( w = y /\ [ w / y ] [ z / x ] ph ) ) )
4		sbcom2 1904	. . . . . . 7 |- ( [ z / x ] [ w / y ] ph <-> [ w / y ] [ z / x ] ph )
5	4	anbi2i 444	. . . . . 6 |- ( ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) <-> ( ( z = x /\ w = y ) /\ [ w / y ] [ z / x ] ph ) )
6		anass 393	. . . . . 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	exbii 1536	. . . 4 |- ( E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) <-> E. w ( z = x /\ ( w = y /\ [ w / y ] [ z / x ] ph ) ) )
9		2sb5rf.2	. . . . . . 7 |- ( ph -> A. w ph )
10	9	hbsbv 1858	. . . . . 6 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
11	10	sb5rf 1773	. . . . 5 |- ( [ z / x ] ph <-> E. w ( w = y /\ [ w / y ] [ z / x ] ph ) )
12	11	anbi2i 444	. . . 4 |- ( ( z = x /\ [ z / x ] ph ) <-> ( z = x /\ E. w ( w = y /\ [ w / y ] [ z / x ] ph ) ) )
13	3, 8, 12	3bitr4ri 211	. . 3 |- ( ( z = x /\ [ z / x ] ph ) <-> E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) )
14	13	exbii 1536	. 2 |- ( E. z ( z = x /\ [ z / x ] ph ) <-> E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) )
15	2, 14	bitri 182	1 |- ( ph <-> E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)