Theorem faimpex 10433

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 2292	. . . . . 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 965	. . . . 5 |- A.x(ph -> ph)
5	2, 4	jctir 293	. . . 4 |- (A =/= (/) -> (E.x x e. A /\ A.x(ph -> ph)))
6		19.29r 1074	. . . 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 1653	. . 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 1762	. 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 956   e. wcel 960  E.wex 982   =/= wne 1588  A.wral 1648  E.wrex 1649  (/)c0 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-nul 2284

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