Theorem immo 1410

Description: "At most one" is preserved through implication (notice wff reversal).

Assertion

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

Proof of Theorem immo

Step	Hyp	Ref	Expression
1		imim1 15	. . . 4 |- ((ph -> ps) -> ((ps -> x = y) -> (ph -> x = y)))
2	1	19.20ii 992	. . 3 |- (A.x(ph -> ps) -> (A.x(ps -> x = y) -> A.x(ph -> x = y)))
3	2	19.22dv 1285	. 2 |- (A.x(ph -> ps) -> (E.yA.x(ps -> x = y) -> E.yA.x(ph -> x = y)))
4		ax-17 968	. . 3 |- (ps -> A.yps)
5	4	mo2 1393	. 2 |- (E*xps <-> E.yA.x(ps -> x = y))
6		ax-17 968	. . 3 |- (ph -> A.yph)
7	6	mo2 1393	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
8	3, 5, 7	3imtr4g 551	1 |- (A.x(ph -> ps) -> (E*xps -> E*xph))

Syntax hints:   -> wi 3  A.wal 951  E.wex 977  E*wmo 1374

This theorem is referenced by:  immoi 1411  euimmo 1413  moexex 1431  brdom6disj 4777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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