Theorem 2alsraln0idm 17067

Description: Nested general "all some" quantifiers with class membership as their antecedents, for the same class A: ph holds for every x and every y in A, and A is inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
2alsraln0idm	|- ( A.E. x ( x e. A -> A.E. y ( y e. A -> ph ) ) <-> ( A. x e. A A. y e. A ph /\ E. x x e. A ) )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem 2alsraln0idm

Step	Hyp	Ref	Expression
1		2alsraln0m 17066	. 2 |- ( A.E. x ( x e. A -> A.E. y ( y e. A -> ph ) ) <-> ( A. x e. A A. y e. A ph /\ ( E. x x e. A /\ E. y y e. A ) ) )
2		eleq1w 2299	. . . . . 6 |- ( y = x -> ( y e. A <-> x e. A ) )
3	2	cbvexvw 1976	. . . . 5 |- ( E. y y e. A <-> E. x x e. A )
4	3	anbi2i 461	. . . 4 |- ( ( E. x x e. A /\ E. y y e. A ) <-> ( E. x x e. A /\ E. x x e. A ) )
5		anidm 400	. . . 4 |- ( ( E. x x e. A /\ E. x x e. A ) <-> E. x x e. A )
6	4, 5	bitri 184	. . 3 |- ( ( E. x x e. A /\ E. y y e. A ) <-> E. x x e. A )
7	6	anbi2i 461	. 2 |- ( ( A. x e. A A. y e. A ph /\ ( E. x x e. A /\ E. y y e. A ) ) <-> ( A. x e. A A. y e. A ph /\ E. x x e. A ) )
8	1, 7	bitri 184	1 |- ( A.E. x ( x e. A -> A.E. y ( y e. A -> ph ) ) <-> ( A. x e. A A. y e. A ph /\ E. x x e. A ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   A.wral 2528   A.E.wals 17034

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-cleq 2231  df-clel 2234  df-ral 2533  df-als 17036

This theorem is referenced by: (None)