Theorem mo4f 2236

Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Hypotheses

Ref	Expression
mo4f.1	|- F/xps
mo4f.2	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
mo4f	|- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

Distinct variable groups:   x,y   ph,y

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

Proof of Theorem mo4f

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 |- F/yph
2	1	mo3 2235	. 2 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
3		mo4f.1	. . . . . 6 |- F/xps
4		mo4f.2	. . . . . 6 |- (x = y -> (ph <-> ps))
5	3, 4	sbie 2038	. . . . 5 |- ([y / x]ph <-> ps)
6	5	anbi2i 675	. . . 4 |- ((ph /\ [y / x]ph) <-> (ph /\ ps))
7	6	imbi1i 315	. . 3 |- (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = y))
8	7	2albii 1567	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) <-> A.xA.y((ph /\ ps) -> x = y))
9	2, 8	bitri 240	1 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544  [wsb 1648  E*wmo 2205

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  mo4  2237  mob2  3016