Theorem immo 1415

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 993	. . 3 |- (A.x(ph -> ps) -> (A.x(ps -> x = y) -> A.x(ph -> x = y)))
3	2	19.22dv 1288	. 2 |- (A.x(ph -> ps) -> (E.yA.x(ps -> x = y) -> E.yA.x(ph -> x = y)))
4		ax-17 969	. . 3 |- (ps -> A.yps)
5	4	mo2 1398	. 2 |- (E*xps <-> E.yA.x(ps -> x = y))
6		ax-17 969	. . 3 |- (ph -> A.yph)
7	6	mo2 1398	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
8	3, 5, 7	3imtr4g 552	1 |- (A.x(ph -> ps) -> (E*xps -> E*xph))

Syntax hints:   -> wi 3  A.wal 952  E.wex 978  E*wmo 1379

This theorem is referenced by:  immoi 1416  euimmo 1418  moexex 1436  brdom6disj 4785

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

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