Theorem mo4 2103

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 1539	. 2 |- F/ x ps
2		mo4.1	. 2 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	mo4f 2102	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E*wmo 2043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046

This theorem is referenced by:  eu4  2104  rmo4  2954  dffun5r  5267  dffun6f  5268  fun11  5322  brprcneu  5548  dff13  5812  mpofun  6021  caovimo  6114  th3qlem1  6693  exmidmotap  7323  addnq0mo  7509  mulnq0mo  7510  addsrmo  7805  mulsrmo  7806  summodc  11529  prodmodc  11724  limcimo  14844