Theorem rr19.3v 2733

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 170	. . . 4 |- ( y = x -> ( ph <-> ph ) )
2	1	rspcv 2697	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2424	. 2 |- ( A. x e. A A. y e. A ph -> A. x e. A ph )
4		ax-1 5	. . . 4 |- ( ph -> ( y e. A -> ph ) )
5	4	ralrimiv 2433	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2426	. 2 |- ( A. x e. A ph -> A. x e. A A. y e. A ph )
7	3, 6	impbii 124	1 |- ( A. x e. A A. y e. A ph <-> A. x e. A ph )

Syntax hints:   <-> wb 103   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603

This theorem is referenced by: (None)