Theorem ralanid 34010

Description: Cancellation law for restriction. (Contributed by Peter Mazsa, 30-Dec-2018.)

Assertion

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

Proof of Theorem ralanid

Step	Hyp	Ref	Expression
1		anclb 570	. . 3 |- ( ( x e. A -> ph ) <-> ( x e. A -> ( x e. A /\ ph ) ) )
2	1	albii 1747	. 2 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. A -> ( x e. A /\ ph ) ) )
3		df-ral 2917	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2917	. 2 |- ( A. x e. A ( x e. A /\ ph ) <-> A. x ( x e. A -> ( x e. A /\ ph ) ) )
5	2, 3, 4	3bitr4ri 293	1 |- ( A. x e. A ( x e. A /\ ph ) <-> A. x e. A ph )

Syntax hints:   -> 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:  idinxpssinxp2  34089