Theorem intmin4 2613

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 2604	. . . 4 |- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
2		pm3.27 330	. . . . . . . 8 |- ((A (_ x /\ ph) -> ph)
3		ancr 302	. . . . . . . 8 |- ((ph -> A (_ x) -> (ph -> (A (_ x /\ ph)))
4	2, 3	impbid2 529	. . . . . . 7 |- ((ph -> A (_ x) -> ((A (_ x /\ ph) <-> ph))
5	4	imbi1d 624	. . . . . 6 |- ((ph -> A (_ x) -> (((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
6	5	19.20i 1033	. . . . 5 |- (A.x(ph -> A (_ x) -> A.x(((A (_ x /\ ph) -> y e. x) <-> (ph -> y e. x)))
7		19.15 1038	. . . . 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 206	. . 3 |- (A (_ |^|{x | ph} -> (A.x((A (_ x /\ ph) -> y e. x) <-> A.x(ph -> y e. x)))
10		visset 1860	. . . 4 |- y e. V
11	10	elintab 2598	. . 3 |- (y e. |^|{x | (A (_ x /\ ph)} <-> A.x((A (_ x /\ ph) -> y e. x))
12	10	elintab 2598	. . 3 |- (y e. |^|{x | ph} <-> A.x(ph -> y e. x))
13	9, 11, 12	3bitr4g 566	. 2 |- (A (_ |^|{x | ph} -> (y e. |^|{x | (A (_ x /\ ph)} <-> y e. |^|{x | ph}))
14	13	eqrdv 1520	1 |- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997   e. wcel 999  {cab 1509   (_ wss 2098  |^|cint 2587

This theorem is referenced by:  abfii3 4623

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ral 1696  df-v 1859  df-in 2102  df-ss 2104  df-int 2588

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