Theorem moim 1961

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moim	|- (A.x(ph -> ps) -> (E*xps -> E*xph))

Proof of Theorem moim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1431	. . 3 |- F/xA.x(ph -> ps)
2		ax-4 1397	. . . . . 6 |- (A.x(ph -> ps) -> (ph -> ps))
3		spsbim 1721	. . . . . 6 |- (A.x(ph -> ps) -> ([y / x]ph -> [y / x]ps))
4	2, 3	anim12d 318	. . . . 5 |- (A.x(ph -> ps) -> ((ph /\ [y / x]ph) -> (ps /\ [y / x]ps)))
5	4	imim1d 69	. . . 4 |- (A.x(ph -> ps) -> (((ps /\ [y / x]ps) -> x = y) -> ((ph /\ [y / x]ph) -> x = y)))
6	5	alimdv 1756	. . 3 |- (A.x(ph -> ps) -> (A.y((ps /\ [y / x]ps) -> x = y) -> A.y((ph /\ [y / x]ph) -> x = y)))
7	1, 6	alimd 1411	. 2 |- (A.x(ph -> ps) -> (A.xA.y((ps /\ [y / x]ps) -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y)))
8		ax-17 1416	. . 3 |- (ps -> A.yps)
9	8	mo3h 1950	. 2 |- (E*xps <-> A.xA.y((ps /\ [y / x]ps) -> x = y))
10		ax-17 1416	. . 3 |- (ph -> A.yph)
11	10	mo3h 1950	. 2 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
12	7, 9, 11	3imtr4g 194	1 |- (A.x(ph -> ps) -> (E*xps -> E*xph))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240  [wsb 1642  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:  moimi  1962  euimmo  1964  moexexdc  1981  euexex  1982  rmoim  2734  rmoimi2  2736  disjss1  3742  reusv1  4156  funmo  4860