Theorem exintrbi 1818

Description: Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Assertion

Ref	Expression
exintrbi	|- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )

Proof of Theorem exintrbi

Step	Hyp	Ref	Expression
1		abai 836	. . 3 |- ( ( ph /\ ps ) <-> ( ph /\ ( ph -> ps ) ) )
2	1	rbaibr 946	. 2 |- ( ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
3	2	alexbii 1760	1 |- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

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-ex 1705

This theorem is referenced by:  exintr  1819