Theorem intmin4 2563

Description: Elimination of a conjunct in a class intersection.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem intmin4

Step	Hyp	Ref	Expression
1		ssintab 2554	. . . 4 |- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
2		pm3.27 323	. . . . . . . 8 |- ((A (_ x /\ ph) -> ph)
3		ancr 295	. . . . . . . 8 |- ((ph -> A (_ x) -> (ph -> (A (_ x /\ ph)))
4	2, 3	impbid2 520	. . . . . . 7 |- ((ph -> A (_ x) -> ((A (_ x /\ ph) <-> ph))
5	4	imbi1d 615	. . . . . 6 |- ((ph -> A (_ x) -> (((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
6	5	19.20i 994	. . . . 5 |- (A.x(ph -> A (_ x) -> A.x(((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
7		19.15 999	. . . . 5 |- (A.x(((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)) -> (A.x((A (_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
8	6, 7	syl 10	. . . 4 |- (A.x(ph -> A (_ x) -> (A.x((A (_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
9	1, 8	sylbi 199	. . 3 |- (A (_ |^|{x | ph} -> (A.x((A (_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
10		visset 1816	. . . 4 |- y e. V
11	10	elintab 2548	. . 3 |- (y e. |^|{x | (A (_ x /\ ph)} <-> A.x((A (_ x /\ ph) -> y e. x))
12	10	elintab 2548	. . 3 |- (y e. |^|{x | ph} <-> A.x(ph -> y e. x))
13	9, 11, 12	3bitr4g 557	. 2 |- (A (_ |^|{x | ph} -> (y e. |^|{x | (A (_ x /\ ph)} <-> y e. |^|{x | ph}))
14	13	eqrdv 1476	1 |- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   (_ wss 2050  |^|cint 2537

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-int 2538

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