Theorem 2ralor 2780

Description: Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)

Assertion

Ref	Expression
2ralor	|- (A.x e. A A.y e. B (ph \/ ps) <-> (A.x e. A ph \/ A.y e. B ps))

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

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

Proof of Theorem 2ralor

Step	Hyp	Ref	Expression
1		rexnal 2625	. . . 4 |- (E.x e. A -. ph <-> -. A.x e. A ph)
2		rexnal 2625	. . . 4 |- (E.y e. B -. ps <-> -. A.y e. B ps)
3	1, 2	anbi12i 678	. . 3 |- ((E.x e. A -. ph /\ E.y e. B -. ps) <-> (-. A.x e. A ph /\ -. A.y e. B ps))
4		ioran 476	. . . . . . 7 |- (-. (ph \/ ps) <-> (-. ph /\ -. ps))
5	4	rexbii 2639	. . . . . 6 |- (E.y e. B -. (ph \/ ps) <-> E.y e. B (-. ph /\ -. ps))
6		rexnal 2625	. . . . . 6 |- (E.y e. B -. (ph \/ ps) <-> -. A.y e. B (ph \/ ps))
7	5, 6	bitr3i 242	. . . . 5 |- (E.y e. B (-. ph /\ -. ps) <-> -. A.y e. B (ph \/ ps))
8	7	rexbii 2639	. . . 4 |- (E.x e. A E.y e. B (-. ph /\ -. ps) <-> E.x e. A -. A.y e. B (ph \/ ps))
9		reeanv 2778	. . . 4 |- (E.x e. A E.y e. B (-. ph /\ -. ps) <-> (E.x e. A -. ph /\ E.y e. B -. ps))
10		rexnal 2625	. . . 4 |- (E.x e. A -. A.y e. B (ph \/ ps) <-> -. A.x e. A A.y e. B (ph \/ ps))
11	8, 9, 10	3bitr3ri 267	. . 3 |- (-. A.x e. A A.y e. B (ph \/ ps) <-> (E.x e. A -. ph /\ E.y e. B -. ps))
12		ioran 476	. . 3 |- (-. (A.x e. A ph \/ A.y e. B ps) <-> (-. A.x e. A ph /\ -. A.y e. B ps))
13	3, 11, 12	3bitr4i 268	. 2 |- (-. A.x e. A A.y e. B (ph \/ ps) <-> -. (A.x e. A ph \/ A.y e. B ps))
14	13	con4bii 288	1 |- (A.x e. A A.y e. B (ph \/ ps) <-> (A.x e. A ph \/ A.y e. B ps))

Syntax hints:  -. wn 3   <-> wb 176   \/ wo 357   /\ wa 358  A.wral 2614  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620

This theorem is referenced by: (None)