Theorem eu4 1412

Description: Uniqueness using implicit substitution.

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

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		eu5 1411	. 2 |- (E!xph <-> (E.xph /\ E*xph))
2		eu4.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	mo4 1405	. . 3 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
4	3	anbi2i 482	. 2 |- ((E.xph /\ E*xph) <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))
5	1, 4	bitr 173	1 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958  E.wex 982  E!weu 1382  E*wmo 1383

This theorem is referenced by:  eueq 1919  climeu 7100  hlimeu 9106

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385

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