Theorem unab 3404

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

Assertion

Ref	Expression
unab	|- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Proof of Theorem unab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbor 1954	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
2		df-clab 2164	. . 3 |- ( y e. { x | ( ph \/ ps ) } <-> [ y / x ] ( ph \/ ps ) )
3		df-clab 2164	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2164	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	orbi12i 764	. . 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	uneqri 3279	1 |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Syntax hints:   \/ wo 708   = wceq 1353   [wsb 1762   e. wcel 2148   {cab 2163   u. cun 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135

This theorem is referenced by:  unrab  3408  rabun2  3416  dfif6  3538  unopab  4084  dmun  4836  frecabex  6401