Theorem rmoan 3035

Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmoan	|- (E*x e. A ph -> E*x e. A (ps /\ ph))

Proof of Theorem rmoan

Step	Hyp	Ref	Expression
1		moan 2255	. . 3 |- (E*x(x e. A /\ ph) -> E*x(ps /\ (x e. A /\ ph)))
2		an12 772	. . . 4 |- ((ps /\ (x e. A /\ ph)) <-> (x e. A /\ (ps /\ ph)))
3	2	mobii 2240	. . 3 |- (E*x(ps /\ (x e. A /\ ph)) <-> E*x(x e. A /\ (ps /\ ph)))
4	1, 3	sylib 188	. 2 |- (E*x(x e. A /\ ph) -> E*x(x e. A /\ (ps /\ ph)))
5		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
6		df-rmo 2623	. 2 |- (E*x e. A (ps /\ ph) <-> E*x(x e. A /\ (ps /\ ph)))
7	4, 5, 6	3imtr4i 257	1 |- (E*x e. A ph -> E*x e. A (ps /\ ph))

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  E*wmo 2205  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-rmo 2623

This theorem is referenced by: (None)