Theorem 2mos 2552

Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)

Hypothesis

Ref	Expression
2mos.1	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
2mos	|- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )

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

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

Proof of Theorem 2mos

Step	Hyp	Ref	Expression
1		2mo 2551	. 2 |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) )
2		nfv 1843	. . . . . . 7 |- F/ x ps
3		2mos.1	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
4	3	sbiedv 2410	. . . . . . 7 |- ( x = z -> ( [ w / y ] ph <-> ps ) )
5	2, 4	sbie 2408	. . . . . 6 |- ( [ z / x ] [ w / y ] ph <-> ps )
6	5	anbi2i 730	. . . . 5 |- ( ( ph /\ [ z / x ] [ w / y ] ph ) <-> ( ph /\ ps ) )
7	6	imbi1i 339	. . . 4 |- ( ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
8	7	2albii 1748	. . 3 |- ( A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
9	8	2albii 1748	. 2 |- ( A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
10	1, 9	bitri 264	1 |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)