Theorem faimpex 10438

Description: "Restricted for all" implies "restricted there exists".

Assertion

Ref	Expression
faimpex	|- (A =/= (/) -> (A.x e. A ph -> E.x e. A ph))

Distinct variable group:   x,A

Proof of Theorem faimpex

Step	Hyp	Ref	Expression
1		ne0 2288	. . . . . 6 |- (A =/= (/) <-> E.x x e. A)
2	1	biimp 151	. . . . 5 |- (A =/= (/) -> E.x x e. A)
3		id 59	. . . . . 6 |- (ph -> ph)
4	3	ax-gen 963	. . . . 5 |- A.x(ph -> ph)
5	2, 4	jctir 293	. . . 4 |- (A =/= (/) -> (E.x x e. A /\ A.x(ph -> ph)))
6		19.29r 1072	. . . 4 |- ((E.x x e. A /\ A.x(ph -> ph)) -> E.x(x e. A /\ (ph -> ph)))
7	5, 6	syl 10	. . 3 |- (A =/= (/) -> E.x(x e. A /\ (ph -> ph)))
8		df-rex 1650	. . 3 |- (E.x e. A (ph -> ph) <-> E.x(x e. A /\ (ph -> ph)))
9	7, 8	sylibr 200	. 2 |- (A =/= (/) -> E.x e. A (ph -> ph))
10		r19.35 1759	. 2 |- (E.x e. A (ph -> ph) <-> (A.x e. A ph -> E.x e. A ph))
11	9, 10	sylib 198	1 |- (A =/= (/) -> (A.x e. A ph -> E.x e. A ph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-nul 2281

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