Theorem 2sb6 2444

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 2429	. 2 |- ( [ z / x ] [ w / y ] ph <-> A. x ( x = z -> [ w / y ] ph ) )
2		19.21v 1868	. . . 4 |- ( A. y ( x = z -> ( y = w -> ph ) ) <-> ( x = z -> A. y ( y = w -> ph ) ) )
3		impexp 462	. . . . 5 |- ( ( ( x = z /\ y = w ) -> ph ) <-> ( x = z -> ( y = w -> ph ) ) )
4	3	albii 1747	. . . 4 |- ( A. y ( ( x = z /\ y = w ) -> ph ) <-> A. y ( x = z -> ( y = w -> ph ) ) )
5		sb6 2429	. . . . 5 |- ( [ w / y ] ph <-> A. y ( y = w -> ph ) )
6	5	imbi2i 326	. . . 4 |- ( ( x = z -> [ w / y ] ph ) <-> ( x = z -> A. y ( y = w -> ph ) ) )
7	2, 4, 6	3bitr4ri 293	. . 3 |- ( ( x = z -> [ w / y ] ph ) <-> A. y ( ( x = z /\ y = w ) -> ph ) )
8	7	albii 1747	. 2 |- ( A. x ( x = z -> [ w / y ] ph ) <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )
9	1, 8	bitri 264	1 |- ( [ z / x ] [ w / y ] ph <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbcom2  2445  2exsb  2469  2eu6  2558