Theorem unab 3476

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 2007	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
2		df-clab 2218	. . 3 |- ( y e. { x | ( ph \/ ps ) } <-> [ y / x ] ( ph \/ ps ) )
3		df-clab 2218	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2218	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	orbi12i 772	. . 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 3351	1 |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Syntax hints:   \/ wo 716   = wceq 1398   [wsb 1810   e. wcel 2202   {cab 2217   u. cun 3199

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205

This theorem is referenced by:  unrab  3480  rabun2  3488  dfif6  3609  unopab  4173  dmun  4944  frecabex  6607  fngsum  13534  igsumvalx  13535  vtxdfifiun  16221