Theorem mo4 1961

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 1421	. 2 |- F/ x ps
2		mo4.1	. 2 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	mo4f 1960	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  eu4  1962  rmo4  2734  dffun5r  4914  dffun6f  4915  fun11  4966  brprcneu  5171  dff13  5407  mpt2fun  5603  caovimo  5694  th3qlem1  6208  addnq0mo  6545  mulnq0mo  6546  addsrmo  6828  mulsrmo  6829