Theorem rr19.28v 2677

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (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 102	. . . . . 6 |- ((ph /\ ps) -> ph)
2	1	ralimi 2378	. . . . 5 |- (A.y e. A (ph /\ ps) -> A.y e. A ph)
3		biidd 161	. . . . . 6 |- (y = x -> (ph <-> ph))
4	3	rspcv 2646	. . . . 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 103	. . . . . 6 |- ((ph /\ ps) -> ps)
7	6	ralimi 2378	. . . . 5 |- (A.y e. A (ph /\ ps) -> A.y e. A ps)
8	7	a1i 9	. . . 4 |- (x e. A -> (A.y e. A (ph /\ ps) -> A.y e. A ps))
9	5, 8	jcad 291	. . 3 |- (x e. A -> (A.y e. A (ph /\ ps) -> (ph /\ A.y e. A ps)))
10	9	ralimia 2376	. 2 |- (A.x e. A A.y e. A (ph /\ ps) -> A.x e. A (ph /\ A.y e. A ps))
11		r19.28av 2443	. . 3 |- ((ph /\ A.y e. A ps) -> A.y e. A (ph /\ ps))
12	11	ralimi 2378	. 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 117	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   /\ wa 97   <-> wb 98   e. wcel 1390  A.wral 2300

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  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553

This theorem is referenced by: (None)