Theorem rr19.28v 3346

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4066 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Assertion

Ref	Expression
rr19.28v	|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> A. x e. A ( ph /\ A. y e. A ps ) )

Distinct variable groups:   y, A   x, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)   A( x)

Proof of Theorem rr19.28v

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
2	1	ralimi 2952	. . . . 5 |- ( A. y e. A ( ph /\ ps ) -> A. y e. A ph )
3		biidd 252	. . . . . 6 |- ( y = x -> ( ph <-> ph ) )
4	3	rspcv 3305	. . . . 5 |- ( x e. A -> ( A. y e. A ph -> ph ) )
5	2, 4	syl5 34	. . . 4 |- ( x e. A -> ( A. y e. A ( ph /\ ps ) -> ph ) )
6		simpr 477	. . . . . 6 |- ( ( ph /\ ps ) -> ps )
7	6	ralimi 2952	. . . . 5 |- ( A. y e. A ( ph /\ ps ) -> A. y e. A ps )
8	7	a1i 11	. . . 4 |- ( x e. A -> ( A. y e. A ( ph /\ ps ) -> A. y e. A ps ) )
9	5, 8	jcad 555	. . 3 |- ( x e. A -> ( A. y e. A ( ph /\ ps ) -> ( ph /\ A. y e. A ps ) ) )
10	9	ralimia 2950	. 2 |- ( A. x e. A A. y e. A ( ph /\ ps ) -> A. x e. A ( ph /\ A. y e. A ps ) )
11		r19.28v 3071	. . 3 |- ( ( ph /\ A. y e. A ps ) -> A. y e. A ( ph /\ ps ) )
12	11	ralimi 2952	. 2 |- ( A. x e. A ( ph /\ A. y e. A ps ) -> A. x e. A A. y e. A ( ph /\ ps ) )
13	10, 12	impbii 199	1 |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> A. x e. A ( ph /\ A. y e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by: (None)