Theorem 2mos 2283

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.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.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 2282	. 2 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
2		nfv 1619	. . . . . . 7 |- F/xps
3		nfv 1619	. . . . . . . . . 10 |- F/y x = z
4	3	sbrim 2067	. . . . . . . . 9 |- ([w / y](x = z -> ph) <-> (x = z -> [w / y]ph))
5		nfv 1619	. . . . . . . . . 10 |- F/y(x = z -> ps)
6		2mos.1	. . . . . . . . . . . 12 |- ((x = z /\ y = w) -> (ph <-> ps))
7	6	expcom 424	. . . . . . . . . . 11 |- (y = w -> (x = z -> (ph <-> ps)))
8	7	pm5.74d 238	. . . . . . . . . 10 |- (y = w -> ((x = z -> ph) <-> (x = z -> ps)))
9	5, 8	sbie 2038	. . . . . . . . 9 |- ([w / y](x = z -> ph) <-> (x = z -> ps))
10	4, 9	bitr3i 242	. . . . . . . 8 |- ((x = z -> [w / y]ph) <-> (x = z -> ps))
11	10	pm5.74ri 237	. . . . . . 7 |- (x = z -> ([w / y]ph <-> ps))
12	2, 11	sbie 2038	. . . . . 6 |- ([z / x][w / y]ph <-> ps)
13	12	anbi2i 675	. . . . 5 |- ((ph /\ [z / x][w / y]ph) <-> (ph /\ ps))
14	13	imbi1i 315	. . . 4 |- (((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> ((ph /\ ps) -> (x = z /\ y = w)))
15	14	2albii 1567	. . 3 |- (A.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.zA.w((ph /\ ps) -> (x = z /\ y = w)))
16	15	2albii 1567	. 2 |- (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))
17	1, 16	bitri 240	1 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)