Theorem rmoim 2734

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
rmoim	|- (A.x e. A (ph -> ps) -> (E*x e. A ps -> E*x e. A ph))

Proof of Theorem rmoim

Step	Hyp	Ref	Expression
1		df-ral 2305	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		imdistan 418	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
3	2	albii 1356	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
4	1, 3	bitri 173	. 2 |- (A.x e. A (ph -> ps) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
5		moim 1961	. . 3 |- (A.x((x e. A /\ ph) -> (x e. A /\ ps)) -> (E*x(x e. A /\ ps) -> E*x(x e. A /\ ph)))
6		df-rmo 2308	. . 3 |- (E*x e. A ps <-> E*x(x e. A /\ ps))
7		df-rmo 2308	. . 3 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
8	5, 6, 7	3imtr4g 194	. 2 |- (A.x((x e. A /\ ph) -> (x e. A /\ ps)) -> (E*x e. A ps -> E*x e. A ph))
9	4, 8	sylbi 114	1 |- (A.x e. A (ph -> ps) -> (E*x e. A ps -> E*x e. A ph))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   e. wcel 1390  E*wmo 1898  A.wral 2300  E*wrmo 2303

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-bndl 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  df-ral 2305  df-rmo 2308

This theorem is referenced by:  rmoimia  2735  disjss2  3739