Theorem unab 3474

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 771	. . 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 3349	1 |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Syntax hints:   \/ wo 715   = wceq 1397   [wsb 1810   e. wcel 2202   {cab 2217   u. cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204

This theorem is referenced by:  unrab  3478  rabun2  3486  dfif6  3607  unopab  4168  dmun  4938  frecabex  6563  fngsum  13470  igsumvalx  13471  vtxdfifiun  16147