Theorem inab 3427

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 1971	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
2		df-clab 2180	. . 3 |- ( y e. { x | ( ph /\ ps ) } <-> [ y / x ] ( ph /\ ps ) )
3		df-clab 2180	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2180	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	anbi12i 460	. . 3 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
6	1, 2, 5	3bitr4ri 213	. 2 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> y e. { x | ( ph /\ ps ) } )
7	6	ineqri 3352	1 |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Syntax hints:   /\ wa 104   = wceq 1364   [wsb 1773   e. wcel 2164   {cab 2179   i^i cin 3152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  inrab  3431  inrab2  3432  dfrab2  3434  dfrab3  3435  imainlem  5335  imain  5336  ssenen  6907