Theorem 2sb6 1908

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

Assertion

Ref	Expression
2sb6	|- ( [ z / x ] [ w / y ] ph <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )

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

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

Proof of Theorem 2sb6

Step	Hyp	Ref	Expression
1		sb6 1814	. 2 |- ( [ z / x ] [ w / y ] ph <-> A. x ( x = z -> [ w / y ] ph ) )
2		19.21v 1801	. . . 4 |- ( A. y ( x = z -> ( y = w -> ph ) ) <-> ( x = z -> A. y ( y = w -> ph ) ) )
3		impexp 259	. . . . 5 |- ( ( ( x = z /\ y = w ) -> ph ) <-> ( x = z -> ( y = w -> ph ) ) )
4	3	albii 1404	. . . 4 |- ( A. y ( ( x = z /\ y = w ) -> ph ) <-> A. y ( x = z -> ( y = w -> ph ) ) )
5		sb6 1814	. . . . 5 |- ( [ w / y ] ph <-> A. y ( y = w -> ph ) )
6	5	imbi2i 224	. . . 4 |- ( ( x = z -> [ w / y ] ph ) <-> ( x = z -> A. y ( y = w -> ph ) ) )
7	2, 4, 6	3bitr4ri 211	. . 3 |- ( ( x = z -> [ w / y ] ph ) <-> A. y ( ( x = z /\ y = w ) -> ph ) )
8	7	albii 1404	. 2 |- ( A. x ( x = z -> [ w / y ] ph ) <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )
9	1, 8	bitri 182	1 |- ( [ z / x ] [ w / y ] ph <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   [wsb 1692

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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-sb 1693

This theorem is referenced by: (None)