Theorem unab 3426

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 1970	. . 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	orbi12i 765	. . 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 3301	1 |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }

Syntax hints:   \/ wo 709   = wceq 1364   [wsb 1773   e. wcel 2164   {cab 2179   u. cun 3151

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-un 3157

This theorem is referenced by:  unrab  3430  rabun2  3438  dfif6  3559  unopab  4108  dmun  4869  frecabex  6451  fngsum  12971  igsumvalx  12972