Theorem mo4 2075

Description: "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem mo4

Step	Hyp	Ref	Expression
1		nfv 1516	. 2 |- F/ x ps
2		mo4.1	. 2 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	mo4f 2074	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  eu4  2076  rmo4  2919  dffun5r  5200  dffun6f  5201  fun11  5255  brprcneu  5479  dff13  5736  mpofun  5944  caovimo  6035  th3qlem1  6603  addnq0mo  7388  mulnq0mo  7389  addsrmo  7684  mulsrmo  7685  summodc  11324  prodmodc  11519  limcimo  13274