Theorem bnj9OLD 12373

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj9OLD	|- ((A.x(ph -> ps) /\ E.x -. ps) -> E.x -. ph)

Proof of Theorem bnj9OLD

Step	Hyp	Ref	Expression
1		ax-5 1302	. . 3 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
2		con3 110	. . . 4 |- ((A.xph -> A.xps) -> (-. A.xps -> -. A.xph))
3		exnal 1385	. . . 4 |- (E.x -. ps <-> -. A.xps)
4		exnal 1385	. . . 4 |- (E.x -. ph <-> -. A.xph)
5	2, 3, 4	3imtr4g 612	. . 3 |- ((A.xph -> A.xps) -> (E.x -. ps -> E.x -. ph))
6	1, 5	syl 12	. 2 |- (A.x(ph -> ps) -> (E.x -. ps -> E.x -. ph))
7	6	imp 377	1 |- ((A.x(ph -> ps) /\ E.x -. ps) -> E.x -. ph)

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

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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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