Theorem mo4 1799

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

Hypothesis

Ref	Expression
mo4.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
mo4	|- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

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

Proof of Theorem mo4

Step	Hyp	Ref	Expression
1		ax-17 1317	. 2 |- (ps -> A.xps)
2		mo4.1	. 2 |- (x = y -> (ph <-> ps))
3	1, 2	mo4f 1798	1 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E*wmo 1772

This theorem is referenced by:  eu4 1806  rmo4 2445  dffun3 4432  fun11 4480  dff13 4850  caoprmo 5003  th3qlem1 5373  supmo 5666  ajmoi 9860  spwmo 9999  hausfillim 10303  adjmo 11395  bra11 11679  bnj149 13241  mpt2fun 13843  ufileu 15573  haustlmu 15906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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