Theorem intmin4 3956

Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem intmin4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintab 3944	. . . 4 |- (A C_ |^|{x | ph} <-> A.x(ph -> A C_ x))
2		simpr 447	. . . . . . . 8 |- ((A C_ x /\ ph) -> ph)
3		ancr 532	. . . . . . . 8 |- ((ph -> A C_ x) -> (ph -> (A C_ x /\ ph)))
4	2, 3	impbid2 195	. . . . . . 7 |- ((ph -> A C_ x) -> ((A C_ x /\ ph) <-> ph))
5	4	imbi1d 308	. . . . . 6 |- ((ph -> A C_ x) -> (((A C_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
6	5	alimi 1559	. . . . 5 |- (A.x(ph -> A C_ x) -> A.x(((A C_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
7		albi 1564	. . . . 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 15	. . . 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 187	. . 3 |- (A C_ |^|{x | ph} -> (A.x((A C_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
10		vex 2863	. . . 4 |- y e. _V
11	10	elintab 3938	. . 3 |- (y e. |^|{x | (A C_ x /\ ph)} <-> A.x((A C_ x /\ ph) -> y e. x))
12	10	elintab 3938	. . 3 |- (y e. |^|{x | ph} <-> A.x(ph -> y e. x))
13	9, 11, 12	3bitr4g 279	. 2 |- (A C_ |^|{x | ph} -> (y e. |^|{x | (A C_ x /\ ph)} <-> y e. |^|{x | ph}))
14	13	eqrdv 2351	1 |- (A C_ |^|{x | ph} -> |^|{x | (A C_ x /\ ph)} = |^|{x | ph})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339   C_ wss 3258  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by: (None)