Theorem bnj111OLD 12456

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj111OLD	|- (ph -> (E!x(ph -> ps) <-> E!xps))

Distinct variable group:   ph,x

Proof of Theorem bnj111OLD

Step	Hyp	Ref	Expression
1		euor2 1839	. 2 |- (-. E.x -. ph -> (E!x(-. ph \/ ps) <-> E!xps))
2		ax-17 1317	. . . 4 |- (ph -> A.xph)
3		ax-4 1319	. . . 4 |- (A.xph -> ph)
4	2, 3	impbii 174	. . 3 |- (ph <-> A.xph)
5		alex 1381	. . 3 |- (A.xph <-> -. E.x -. ph)
6	4, 5	bitri 190	. 2 |- (ph <-> -. E.x -. ph)
7		imor 251	. . . 4 |- ((ph -> ps) <-> (-. ph \/ ps))
8	7	eubii 1780	. . 3 |- (E!x(ph -> ps) <-> E!x(-. ph \/ ps))
9	8	bibi1i 671	. 2 |- ((E!x(ph -> ps) <-> E!xps) <-> (E!x(-. ph \/ ps) <-> E!xps))
10	1, 6, 9	3imtr4i 236	1 |- (ph -> (E!x(ph -> ps) <-> E!xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239  A.wal 1296  E.wex 1326  E!weu 1771

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-eu 1775

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