Theorem rr19.3v 2919

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 2880	. . 3 |- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia 2569	. 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 2580	. . 3 |- ( ph -> A. y e. A ph )
6	5	ralimi 2571	. 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 2178   A.wral 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778

This theorem is referenced by: (None)