Theorem ac5g 14388

Description: ac5 5914 with the premisse transformed into an antecedent.

Assertion

Ref	Expression
ac5g	|- (A e. _V -> E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x)))

Distinct variable group:   A,f,x

Proof of Theorem ac5g

Step	Hyp	Ref	Expression
1		fneq2 4504	. . . 4 |- (A = if(A e. _V, A, (/)) -> (f Fn A <-> f Fn if(A e. _V, A, (/))))
2		raleq 2266	. . . 4 |- (A = if(A e. _V, A, (/)) -> (A.x e. A (x =/= (/) -> (f` x) e. x) <-> A.x e. if (A e. _V, A, (/))(x =/= (/) -> (f` x) e. x)))
3	1, 2	anbi12d 690	. . 3 |- (A = if(A e. _V, A, (/)) -> ((f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x)) <-> (f Fn if(A e. _V, A, (/)) /\ A.x e. if (A e. _V, A, (/))(x =/= (/) -> (f` x) e. x))))
4	3	exbidv 1657	. 2 |- (A = if(A e. _V, A, (/)) -> (E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x)) <-> E.f(f Fn if(A e. _V, A, (/)) /\ A.x e. if (A e. _V, A, (/))(x =/= (/) -> (f` x) e. x))))
5		0ex 3446	. . . 4 |- (/) e. _V
6	5	elimel 3025	. . 3 |- if(A e. _V, A, (/)) e. _V
7	6	ac5 5914	. 2 |- E.f(f Fn if(A e. _V, A, (/)) /\ A.x e. if (A e. _V, A, (/))(x =/= (/) -> (f` x) e. x))
8	4, 7	dedth 3011	1 |- (A e. _V -> E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  ifcif 2982   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  osneisi 14875

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-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

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  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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