Theorem intabs 4825

Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   A( y)

Proof of Theorem intabs

Step	Hyp	Ref	Expression
1		sseq1 3626	. . . . . 6 |- ( x = |^| { y | ps } -> ( x C_ A <-> |^| { y | ps } C_ A ) )
2		intabs.2	. . . . . 6 |- ( x = |^| { y | ps } -> ( ph <-> ch ) )
3	1, 2	anbi12d 747	. . . . 5 |- ( x = |^| { y | ps } -> ( ( x C_ A /\ ph ) <-> ( |^| { y | ps } C_ A /\ ch ) ) )
4		intabs.3	. . . . 5 |- ( |^| { y | ps } C_ A /\ ch )
5	3, 4	intmin3 4505	. . . 4 |- ( |^| { y | ps } e. _V -> |^| { x | ( x C_ A /\ ph ) } C_ |^| { y | ps } )
6		intnex 4821	. . . . 5 |- ( -. |^| { y | ps } e. _V <-> |^| { y | ps } = _V )
7		ssv 3625	. . . . . 6 |- |^| { x | ( x C_ A /\ ph ) } C_ _V
8		sseq2 3627	. . . . . 6 |- ( |^| { y | ps } = _V -> ( |^| { x | ( x C_ A /\ ph ) } C_ |^| { y | ps } <-> |^| { x | ( x C_ A /\ ph ) } C_ _V ) )
9	7, 8	mpbiri 248	. . . . 5 |- ( |^| { y | ps } = _V -> |^| { x | ( x C_ A /\ ph ) } C_ |^| { y | ps } )
10	6, 9	sylbi 207	. . . 4 |- ( -. |^| { y | ps } e. _V -> |^| { x | ( x C_ A /\ ph ) } C_ |^| { y | ps } )
11	5, 10	pm2.61i 176	. . 3 |- |^| { x | ( x C_ A /\ ph ) } C_ |^| { y | ps }
12		intabs.1	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
13	12	cbvabv 2747	. . . 4 |- { x | ph } = { y | ps }
14	13	inteqi 4479	. . 3 |- |^| { x | ph } = |^| { y | ps }
15	11, 14	sseqtr4i 3638	. 2 |- |^| { x | ( x C_ A /\ ph ) } C_ |^| { x | ph }
16		simpr 477	. . . 4 |- ( ( x C_ A /\ ph ) -> ph )
17	16	ss2abi 3674	. . 3 |- { x | ( x C_ A /\ ph ) } C_ { x | ph }
18		intss 4498	. . 3 |- ( { x | ( x C_ A /\ ph ) } C_ { x | ph } -> |^| { x | ph } C_ |^| { x | ( x C_ A /\ ph ) } )
19	17, 18	ax-mp 5	. 2 |- |^| { x | ph } C_ |^| { x | ( x C_ A /\ ph ) }
20	15, 19	eqssi 3619	1 |- |^| { x | ( x C_ A /\ ph ) } = |^| { x | ph }

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476

This theorem is referenced by:  dfnn3  11034