Theorem intabs 2701

Description: Absorption of a redundant conjunct in the intersection of a class abstraction.

Hypotheses

Ref	Expression
intabs.1	|- (x = y -> (ph <-> ps))
intabs.2	|- (x = |^|{y | ps} -> (ph <-> ch))
intabs.3	|- (|^|{y | ps} (_ A /\ ch)

Assertion

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

Distinct variable groups:   x,y   x,A   ph,y   ps,x   ch,x

Proof of Theorem intabs

Step	Hyp	Ref	Expression
1		sseq1 2053	. . . . . 6 |- (x = |^|{y | ps} -> (x (_ A <-> |^|{y | ps} (_ A))
2		intabs.2	. . . . . 6 |- (x = |^|{y | ps} -> (ph <-> ch))
3	1, 2	anbi12d 626	. . . . 5 |- (x = |^|{y | ps} -> ((x (_ A /\ ph) <-> (|^|{y | ps} (_ A /\ ch)))
4		intabs.3	. . . . 5 |- (|^|{y | ps} (_ A /\ ch)
5	3, 4	intmin3 2526	. . . 4 |- (|^|{y | ps} e. V -> |^|{x | (x (_ A /\ ph)} (_ |^|{y | ps})
6		intnex 2698	. . . . 5 |- (-. |^|{y | ps} e. V <-> |^|{y | ps} = V)
7		ssv 2052	. . . . . 6 |- |^|{x | (x (_ A /\ ph)} (_ V
8		sseq2 2054	. . . . . 6 |- (|^|{y | ps} = V -> (|^|{x | (x (_ A /\ ph)} (_ |^|{y | ps} <-> |^|{x | (x (_ A /\ ph)} (_ V))
9	7, 8	mpbiri 194	. . . . 5 |- (|^|{y | ps} = V -> |^|{x | (x (_ A /\ ph)} (_ |^|{y | ps})
10	6, 9	sylbi 199	. . . 4 |- (-. |^|{y | ps} e. V -> |^|{x | (x (_ A /\ ph)} (_ |^|{y | ps})
11	5, 10	pm2.61i 126	. . 3 |- |^|{x | (x (_ A /\ ph)} (_ |^|{y | ps}
12		intabs.1	. . . . 5 |- (x = y -> (ph <-> ps))
13	12	cbvabv 1881	. . . 4 |- {x | ph} = {y | ps}
14	13	inteqi 2505	. . 3 |- |^|{x | ph} = |^|{y | ps}
15	11, 14	sseqtr4 2065	. 2 |- |^|{x | (x (_ A /\ ph)} (_ |^|{x | ph}
16		pm3.27 323	. . . 4 |- ((x (_ A /\ ph) -> ph)
17	16	ss2abi 2091	. . 3 |- {x | (x (_ A /\ ph)} (_ {x | ph}
18		intss 2522	. . 3 |- ({x | (x (_ A /\ ph)} (_ {x | ph} -> |^|{x | ph} (_ |^|{x | (x (_ A /\ ph)})
19	17, 18	ax-mp 7	. 2 |- |^|{x | ph} (_ |^|{x | (x (_ A /\ ph)}
20	15, 19	eqssi 2049	1 |- |^|{x | (x (_ A /\ ph)} = |^|{x | ph}

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786   (_ wss 2018  |^|cint 2501

This theorem is referenced by:  dfnn2 5835

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-13 1107  ax-14 1108  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-in 2022  df-ss 2024  df-nul 2252  df-int 2502

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