Theorem wl-alanbii 33351

Description: This theorem extends alanimi 1744 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)

Hypothesis

Ref	Expression
wl-alanbii.1	|- ( ph <-> ( ps /\ ch ) )

Assertion

Ref	Expression
wl-alanbii	|- ( A. x ph <-> ( A. x ps /\ A. x ch ) )

Proof of Theorem wl-alanbii

Step	Hyp	Ref	Expression
1		wl-alanbii.1	. . 3 |- ( ph <-> ( ps /\ ch ) )
2	1	albii 1747	. 2 |- ( A. x ph <-> A. x ( ps /\ ch ) )
3		19.26 1798	. 2 |- ( A. x ( ps /\ ch ) <-> ( A. x ps /\ A. x ch ) )
4	2, 3	bitri 264	1 |- ( A. x ph <-> ( A. x ps /\ A. x ch ) )

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

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

This theorem is referenced by: (None)