Theorem 2sb5 1983

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

Assertion

Ref	Expression
2sb5	|- ( [ z / x ] [ w / y ] ph <-> E. x E. 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 2sb5

Step	Hyp	Ref	Expression
1		sb5 1887	. 2 |- ( [ z / x ] [ w / y ] ph <-> E. x ( x = z /\ [ w / y ] ph ) )
2		19.42v 1906	. . . 4 |- ( E. y ( x = z /\ ( y = w /\ ph ) ) <-> ( x = z /\ E. y ( y = w /\ ph ) ) )
3		anass 401	. . . . 5 |- ( ( ( x = z /\ y = w ) /\ ph ) <-> ( x = z /\ ( y = w /\ ph ) ) )
4	3	exbii 1605	. . . 4 |- ( E. y ( ( x = z /\ y = w ) /\ ph ) <-> E. y ( x = z /\ ( y = w /\ ph ) ) )
5		sb5 1887	. . . . 5 |- ( [ w / y ] ph <-> E. y ( y = w /\ ph ) )
6	5	anbi2i 457	. . . 4 |- ( ( x = z /\ [ w / y ] ph ) <-> ( x = z /\ E. y ( y = w /\ ph ) ) )
7	2, 4, 6	3bitr4ri 213	. . 3 |- ( ( x = z /\ [ w / y ] ph ) <-> E. y ( ( x = z /\ y = w ) /\ ph ) )
8	7	exbii 1605	. 2 |- ( E. x ( x = z /\ [ w / y ] ph ) <-> E. x E. y ( ( x = z /\ y = w ) /\ ph ) )
9	1, 8	bitri 184	1 |- ( [ z / x ] [ w / y ] ph <-> E. x E. y ( ( x = z /\ y = w ) /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1492   [wsb 1762

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  opelopabsbALT  4258