Theorem mo4f 1400

Description: "At most one" expressed using implicit substitution.

Hypotheses

Ref	Expression
mo4f.1	|- (ps -> A.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

Proof of Theorem mo4f

Step	Hyp	Ref	Expression
1		ax-17 969	. . 3 |- (ph -> A.yph)
2	1	mo3 1399	. 2 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
3		mo4f.1	. . . . . 6 |- (ps -> A.xps)
4		mo4f.2	. . . . . 6 |- (x = y -> (ph <-> ps))
5	3, 4	sbie 1194	. . . . 5 |- ([y / x]ph <-> ps)
6	5	anbi2i 480	. . . 4 |- ((ph /\ [y / x]ph) <-> (ph /\ ps))
7	6	imbi1i 186	. . 3 |- (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = y))
8	7	2albii 998	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) <-> A.xA.y((ph /\ ps) -> x = y))
9	2, 8	bitr 173	1 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954  E*wmo 1379

This theorem is referenced by:  mo4 1401  moi2 1920  moi 1921  moop2 2796

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

Copyright terms: Public domain