Theorem moim 2250

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		imim1 70	. . . 4 |- ((ph -> ps) -> ((ps -> x = y) -> (ph -> x = y)))
2	1	al2imi 1561	. . 3 |- (A.x(ph -> ps) -> (A.x(ps -> x = y) -> A.x(ph -> x = y)))
3	2	eximdv 1622	. 2 |- (A.x(ph -> ps) -> (E.yA.x(ps -> x = y) -> E.yA.x(ph -> x = y)))
4		nfv 1619	. . 3 |- F/yps
5	4	mo2 2233	. 2 |- (E*xps <-> E.yA.x(ps -> x = y))
6		nfv 1619	. . 3 |- F/yph
7	6	mo2 2233	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
8	3, 5, 7	3imtr4g 261	1 |- (A.x(ph -> ps) -> (E*xps -> E*xph))

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541   = wceq 1642  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  moimi  2251  euimmo  2253  moexex  2273  rmoim  3035  rmoimi2  3037  funmo  5125