Theorem intmin4 3246

Description: Elimination of a conjunct in a class intersection.

Assertion

Ref	Expression
intmin4	|- (A C_ |^|{x | ph} -> |^|{x | (A C_ x /\ ph)} = |^|{x | ph})

Distinct variable group:   x,A

Proof of Theorem intmin4

Step	Hyp	Ref	Expression
1		ssintab 3233	. . . 4 |- (A C_ |^|{x | ph} <-> A.x(ph -> A C_ x))
2		simpr 350	. . . . . . . 8 |- ((A C_ x /\ ph) -> ph)
3		ancr 319	. . . . . . . 8 |- ((ph -> A C_ x) -> (ph -> (A C_ x /\ ph)))
4	2, 3	impbid2 576	. . . . . . 7 |- ((ph -> A C_ x) -> ((A C_ x /\ ph) <-> ph))
5	4	imbi1d 675	. . . . . 6 |- ((ph -> A C_ x) -> (((A C_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
6	5	alimi 1338	. . . . 5 |- (A.x(ph -> A C_ x) -> A.x(((A C_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
7		albi 1344	. . . . 5 |- (A.x(((A C_ x /\ ph) -> y e. x) <-> (ph -> y e. x)) -> (A.x((A C_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
8	6, 7	syl 12	. . . 4 |- (A.x(ph -> A C_ x) -> (A.x((A C_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
9	1, 8	sylbi 216	. . 3 |- (A C_ |^|{x | ph} -> (A.x((A C_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
10		visset 2295	. . . 4 |- y e. _V
11	10	elintab 3227	. . 3 |- (y e. |^|{x | (A C_ x /\ ph)} <-> A.x((A C_ x /\ ph) -> y e. x))
12	10	elintab 3227	. . 3 |- (y e. |^|{x | ph} <-> A.x(ph -> y e. x))
13	9, 11, 12	3bitr4g 614	. 2 |- (A C_ |^|{x | ph} -> (y e. |^|{x | (A C_ x /\ ph)} <-> y e. |^|{x | ph}))
14	13	eqrdv 1882	1 |- (A C_ |^|{x | ph} -> |^|{x | (A C_ x /\ ph)} = |^|{x | ph})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   C_ wss 2593  |^|cint 3214

This theorem is referenced by:  abfii3 5653

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  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-int 3215

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