Theorem inab 3205

Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inab	|- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Proof of Theorem inab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sban 1829	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
2		df-clab 2027	. . 3 |- ( y e. { x | ( ph /\ ps ) } <-> [ y / x ] ( ph /\ ps ) )
3		df-clab 2027	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2027	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	anbi12i 433	. . 3 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
6	1, 2, 5	3bitr4ri 202	. 2 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> y e. { x | ( ph /\ ps ) } )
7	6	ineqri 3130	1 |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   [wsb 1645   {cab 2026   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924

This theorem is referenced by:  inrab  3209  inrab2  3210  dfrab2  3212  dfrab3  3213  imainlem  4980  imain  4981