Theorem bnj599 12560

Description: First-order logic and set theory. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
bnj599	|- (ph -> (ps -> E!xch))

Proof of Theorem bnj599

Step	Hyp	Ref	Expression
1		bnj599.1	. . 3 |- (ph -> (ps -> E.xch))
2		bnj599.2	. . 3 |- (ph -> (ps -> E*xch))
3	1, 2	jcad 661	. 2 |- (ph -> (ps -> (E.xch /\ E*xch)))
4		eu5 1805	. 2 |- (E!xch <-> (E.xch /\ E*xch))
5	3, 4	syl6ibr 230	1 |- (ph -> (ps -> E!xch))

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

This theorem is referenced by:  bnj600 13308

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

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