Theorem rr19.3v 2876

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 172	. . . 4 |- ( y = x -> ( ph <-> ph ) )
2	1	rspcv 2837	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2538	. 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 2549	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2540	. 2 |- ( A. x e. A ph -> A. x e. A A. y e. A ph )
7	3, 6	impbii 126	1 |- ( A. x e. A A. y e. A ph <-> A. x e. A ph )

Syntax hints:   <-> wb 105   e. wcel 2148   A.wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739

This theorem is referenced by: (None)