Theorem bnj599OLD 12561

Description: First-order logic and set theory.

Hypotheses

Ref	Expression
bnj599.1	|- (ph -> (ps -> E.xch))
bnj599.2	|- (ph -> (ps -> E*xch))

Assertion

Ref	Expression
bnj599OLD	|- (ph -> (ps -> E!xch))

Proof of Theorem bnj599OLD

Step	Hyp	Ref	Expression
1		eu5 1805	. . 3 |- (E!xch <-> (E.xch /\ E*xch))
2		bnj599.1	. . . 4 |- (ph -> (ps -> E.xch))
3	2	imp 377	. . 3 |- ((ph /\ ps) -> E.xch)
4		bnj599.2	. . . 4 |- (ph -> (ps -> E*xch))
5	4	imp 377	. . 3 |- ((ph /\ ps) -> E*xch)
6	1, 3, 5	sylanbrc 527	. 2 |- ((ph /\ ps) -> E!xch)
7	6	ex 402	1 |- (ph -> (ps -> E!xch))

Syntax hints:   -> wi 3   /\ wa 240  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

Copyright terms: Public domain