Theorem mo4 1958

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*xph <-> A.xA.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 1418	. 2 |- F/xps
2		mo4.1	. 2 |- (x = y -> (ph <-> ps))
3	1, 2	mo4f 1957	1 |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E*wmo 1898

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  eu4  1959  rmo4  2728  dffun5r  4857  dffun6f  4858  fun11  4909  brprcneu  5114  dff13  5350  mpt2fun  5545  caovimo  5636  th3qlem1  6144  addnq0mo  6429  mulnq0mo  6430  addsrmo  6651  mulsrmo  6652