Theorem 19.35 1712

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier.

Assertion

Ref	Expression
19.35	|- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		19.26 1703	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 365	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	albii 1635	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		df-an 339	. . . 4 |- ((A.xph /\ A.x -. ps) <-> -. (A.xph -> -. A.x -. ps))
5	1, 3, 4	3bitr3i 293	. . 3 |- (A.x -. (ph -> ps) <-> -. (A.xph -> -. A.x -. ps))
6	5	con2bii 335	. 2 |- ((A.xph -> -. A.x -. ps) <-> -. A.x -. (ph -> ps))
7		df-ex 1616	. . 3 |- (E.xps <-> -. A.x -. ps)
8	7	imbi2i 297	. 2 |- ((A.xph -> E.xps) <-> (A.xph -> -. A.x -. ps))
9		df-ex 1616	. 2 |- (E.x(ph -> ps) <-> -. A.x -. (ph -> ps))
10	6, 8, 9	3bitr4ri 296	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 219   /\ wa 337  A.wal 1584  E.wex 1615

This theorem is referenced by:  19.35i 1713  19.35ri 1714  19.36 1715  19.37 1717  19.39 1719  19.24 1720  19.25 1721  sbequi 1874  grothprim 11012

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1593  ax-4 1608  ax-5o 1610

This theorem depends on definitions:  df-bi 220  df-an 339  df-ex 1616

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