Theorem rr19.3v 2827

Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)

Assertion

Ref	Expression
rr19.3v	|- ( A. x e. A A. y e. A ph <-> A. x e. A ph )

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

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rr19.3v

Step	Hyp	Ref	Expression
1		biidd 171	. . . 4 |- ( y = x -> ( ph <-> ph ) )
2	1	rspcv 2789	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2496	. 2 |- ( A. x e. A A. y e. A ph -> A. x e. A ph )
4		ax-1 6	. . . 4 |- ( ph -> ( y e. A -> ph ) )
5	4	ralrimiv 2507	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2498	. 2 |- ( A. x e. A ph -> A. x e. A A. y e. A ph )
7	3, 6	impbii 125	1 |- ( A. x e. A A. y e. A ph <-> A. x e. A ph )

Syntax hints:   <-> wb 104   e. wcel 1481   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691

This theorem is referenced by: (None)