Theorem rmoimi2 3037

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

Hypothesis

Ref	Expression
rmoimi2.1	|- A.x((x e. A /\ ph) -> (x e. B /\ ps))

Assertion

Ref	Expression
rmoimi2	|- (E*x e. B ps -> E*x e. A ph)

Proof of Theorem rmoimi2

Step	Hyp	Ref	Expression
1		rmoimi2.1	. . 3 |- A.x((x e. A /\ ph) -> (x e. B /\ ps))
2		moim 2250	. . 3 |- (A.x((x e. A /\ ph) -> (x e. B /\ ps)) -> (E*x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
3	1, 2	ax-mp 5	. 2 |- (E*x(x e. B /\ ps) -> E*x(x e. A /\ ph))
4		df-rmo 2622	. 2 |- (E*x e. B ps <-> E*x(x e. B /\ ps))
5		df-rmo 2622	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
6	3, 4, 5	3imtr4i 257	1 |- (E*x e. B ps -> E*x e. A ph)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   e. wcel 1710  E*wmo 2205  E*wrmo 2617

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-rmo 2622

This theorem is referenced by: (None)