Theorem rexals 17064

Description: If some x in A satisfies ph, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that ph holds for every x in A. See rexrals 17058 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)

Assertion

Ref	Expression
rexals	|- ( E. x e. A ph -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexals

Step	Hyp	Ref	Expression
1		alsralrex 17061	. 2 |- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x e. A ph ) )
2		iba 300	. . 3 |- ( E. x e. A ph -> ( A. x e. A ph <-> ( A. x e. A ph /\ E. x e. A ph ) ) )
3	2	bicomd 141	. 2 |- ( E. x e. A ph -> ( ( A. x e. A ph /\ E. x e. A ph ) <-> A. x e. A ph ) )
4	1, 3	bitrid 192	1 |- ( E. x e. A ph -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   A.wral 2528   E.wrex 2529   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

This theorem depends on definitions:  df-bi 117  df-clel 2234  df-ral 2533  df-rex 2534  df-als 17036

This theorem is referenced by: (None)