Theorem eu4 2243

Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

Ref	Expression
eu4.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
eu4	|- (E!xph <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))

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

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

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 |- (E!xph <-> (E.xph /\ E*xph))
2		eu4.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	mo4 2237	. . 3 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
4	3	anbi2i 675	. 2 |- ((E.xph /\ E*xph) <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))
5	1, 4	bitri 240	1 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  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:  euequ1  2292  eueq  3009  euind  3024  uniintsn  3964