Theorem rr19.3v 2769

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 2732	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2447	. 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 2457	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2449	. 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 1445   A.wral 2370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635

This theorem is referenced by: (None)