Theorem rmoim 3036

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 2620	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		imdistan 670	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
3	2	albii 1566	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
4	1, 3	bitri 240	. 2 |- (A.x e. A (ph -> ps) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
5		moim 2250	. . 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 2623	. . 3 |- (E*x e. A ps <-> E*x(x e. A /\ ps))
7		df-rmo 2623	. . 3 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
8	5, 6, 7	3imtr4g 261	. 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 187	1 |- (A.x e. A (ph -> ps) -> (E*x e. A ps -> E*x e. A ph))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   e. wcel 1710  E*wmo 2205  A.wral 2615  E*wrmo 2618

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  df-ral 2620  df-rmo 2623

This theorem is referenced by:  rmoimia  3037  2rmorex  3041