Theorem exintrbi 1647

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		pm4.71 389	. . 3 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
2	1	albii 1484	. 2 |- ( A. x ( ph -> ps ) <-> A. x ( ph <-> ( ph /\ ps ) ) )
3		exbi 1618	. 2 |- ( A. x ( ph <-> ( ph /\ ps ) ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )
4	2, 3	sylbi 121	1 |- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exintr  1648