Theorem rr19.28v 2981

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3645 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Assertion

Ref	Expression
rr19.28v	|- (A.x e. A A.y e. A (ph /\ ps) <-> A.x e. A (ph /\ A.y e. A ps))

Distinct variable groups:   y,A   x,y   ph,y

Allowed substitution hints:   ph(x)   ps(x,y)   A(x)

Proof of Theorem rr19.28v

Step	Hyp	Ref	Expression
1		simpl 443	. . . . . 6 |- ((ph /\ ps) -> ph)
2	1	ralimi 2689	. . . . 5 |- (A.y e. A (ph /\ ps) -> A.y e. A ph)
3		biidd 228	. . . . . 6 |- (y = x -> (ph <-> ph))
4	3	rspcv 2951	. . . . 5 |- (x e. A -> (A.y e. A ph -> ph))
5	2, 4	syl5 28	. . . 4 |- (x e. A -> (A.y e. A (ph /\ ps) -> ph))
6		simpr 447	. . . . . 6 |- ((ph /\ ps) -> ps)
7	6	ralimi 2689	. . . . 5 |- (A.y e. A (ph /\ ps) -> A.y e. A ps)
8	7	a1i 10	. . . 4 |- (x e. A -> (A.y e. A (ph /\ ps) -> A.y e. A ps))
9	5, 8	jcad 519	. . 3 |- (x e. A -> (A.y e. A (ph /\ ps) -> (ph /\ A.y e. A ps)))
10	9	ralimia 2687	. 2 |- (A.x e. A A.y e. A (ph /\ ps) -> A.x e. A (ph /\ A.y e. A ps))
11		r19.28av 2753	. . 3 |- ((ph /\ A.y e. A ps) -> A.y e. A (ph /\ ps))
12	11	ralimi 2689	. 2 |- (A.x e. A (ph /\ A.y e. A ps) -> A.x e. A A.y e. A (ph /\ ps))
13	10, 12	impbii 180	1 |- (A.x e. A A.y e. A (ph /\ ps) <-> A.x e. A (ph /\ A.y e. A ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614

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  ax-ext 2334

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861

This theorem is referenced by: (None)