Theorem exintrbi 1613

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.xph <-> E.x(ph /\ ps)))

Proof of Theorem exintrbi

Step	Hyp	Ref	Expression
1		pm4.71 611	. . 3 |- ((ph -> ps) <-> (ph <-> (ph /\ ps)))
2	1	albii 1566	. 2 |- (A.x(ph -> ps) <-> A.x(ph <-> (ph /\ ps)))
3		exbi 1581	. 2 |- (A.x(ph <-> (ph /\ ps)) -> (E.xph <-> E.x(ph /\ ps)))
4	2, 3	sylbi 187	1 |- (A.x(ph -> ps) -> (E.xph <-> E.x(ph /\ ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  exintr  1614