Theorem 19.9h 1531

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)

Hypothesis

Ref	Expression
19.9h.1	|- (ph -> A.xph)

Assertion

Ref	Expression
19.9h	|- (E.xph <-> ph)

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9ht 1529	. . 3 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
2		19.9h.1	. . 3 |- (ph -> A.xph)
3	1, 2	mpg 1337	. 2 |- (E.xph -> ph)
4		19.8a 1479	. 2 |- (ph -> E.xph)
5	3, 4	impbii 117	1 |- (E.xph <-> ph)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.9  1532  excomim  1550  exdistrfor  1678  sbcof2  1688  ax11ev  1706  19.9v  1748  exists1  1993