Theorem 19.35 1426

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 1416	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 257	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	albii 1346	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		df-an 242	. . . 4 |- ((A.xph /\ A.x -. ps) <-> -. (A.xph -> -. A.x -. ps))
5	1, 3, 4	3bitr3i 198	. . 3 |- (A.x -. (ph -> ps) <-> -. (A.xph -> -. A.x -. ps))
6	5	con2bii 238	. 2 |- ((A.xph -> -. A.x -. ps) <-> -. A.x -. (ph -> ps))
7		df-ex 1327	. . 3 |- (E.xps <-> -. A.x -. ps)
8	7	imbi2i 202	. 2 |- ((A.xph -> E.xps) <-> (A.xph -> -. A.x -. ps))
9		df-ex 1327	. 2 |- (E.x(ph -> ps) <-> -. A.x -. (ph -> ps))
10	6, 8, 9	3bitr4ri 201	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296  E.wex 1326

This theorem is referenced by:  19.35i 1427  19.35ri 1428  19.36 1429  19.37 1431  19.39 1433  19.24 1434  19.25 1435  sbequi 1598  grothprim 10141

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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