Theorem inabs3 39224

Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
inabs3	|- ( C C_ B -> ( ( A i^i B ) i^i C ) = ( A i^i C ) )

Proof of Theorem inabs3

Step	Hyp	Ref	Expression
1		inass 3823	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
2		sseqin2 3817	. . . 4 |- ( C C_ B <-> ( B i^i C ) = C )
3	2	biimpi 206	. . 3 |- ( C C_ B -> ( B i^i C ) = C )
4	3	ineq2d 3814	. 2 |- ( C C_ B -> ( A i^i ( B i^i C ) ) = ( A i^i C ) )
5	1, 4	syl5eq 2668	1 |- ( C C_ B -> ( ( A i^i B ) i^i C ) = ( A i^i C ) )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   C_ wss 3574

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  carageniuncllem1  40735