Theorem unab 3444

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 1983	. . 3 |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) )
2		df-clab 2193	. . 3 |- ( y e. { x | ( ph \/ ps ) } <-> [ y / x ] ( ph \/ ps ) )
3		df-clab 2193	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2193	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	orbi12i 766	. . 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 3319	1 |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Syntax hints:   \/ wo 710   = wceq 1373   [wsb 1786   e. wcel 2177   {cab 2192   u. cun 3168

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174

This theorem is referenced by:  unrab  3448  rabun2  3456  dfif6  3577  unopab  4131  dmun  4894  frecabex  6497  fngsum  13295  igsumvalx  13296