Theorem raln 2991

Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)

Assertion

Ref	Expression
raln	|- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )

Proof of Theorem raln

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
2		imnang 1769	. 2 |- ( A. x ( x e. A -> -. ph ) <-> A. x -. ( x e. A /\ ph ) )
3	1, 2	bitri 264	1 |- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  ralnex  2992  rabeq0  3957