Theorem 19.9 1012

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
19.9.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem 19.9

Step	Hyp	Ref	Expression
1		19.9t 1011	. . 3 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
2		19.9.1	. . 3 |- (ph -> A.xph)
3	1, 2	mpg 962	. 2 |- (E.xph -> ph)
4		19.8a 1005	. 2 |- (ph -> E.xph)
5	3, 4	impbi 157	1 |- (E.xph <-> ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950  E.wex 956

This theorem is referenced by:  excomim 1021  19.19 1031  19.23 1039  19.23ai 1040  19.36 1054  19.44 1065  19.45 1066  19.9v 1266  exists1 1434  dfid3 2799

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955

This theorem depends on definitions:  df-bi 147  df-ex 957

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