Theorem rr19.3v 2869

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 2830	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2531	. 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 2542	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2533	. 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 2141   A.wral 2448

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732

This theorem is referenced by: (None)