Theorem intmin4 2527

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 2518	. . . 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 516	. . . . . . 7 |- ((ph -> A (_ x) -> ((A (_ x /\ ph) <-> ph))
5	4	imbi1d 611	. . . . . 6 |- ((ph -> A (_ x) -> (((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
6	5	19.20i 968	. . . . 5 |- (A.x(ph -> A (_ x) -> A.x(((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
7		19.15 973	. . . . 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 1788	. . . 4 |- y e. V
11	10	elintab 2512	. . 3 |- (y e. |^|{x | (A (_ x /\ ph)} <-> A.x((A (_ x /\ ph) -> y e. x))
12	10	elintab 2512	. . 3 |- (y e. |^|{x | ph} <-> A.x(ph -> y e. x))
13	9, 11, 12	3bitr4g 553	. 2 |- (A (_ |^|{x | ph} -> (y e. |^|{x | (A (_ x /\ ph)} <-> y e. |^|{x | ph}))
14	13	eqrdv 1450	1 |- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  {cab 1440   (_ wss 2018  |^|cint 2501

This theorem is referenced by:  abfii3 4489

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ral 1625  df-v 1787  df-in 2022  df-ss 2024  df-int 2502

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