Theorem rabxm 15667

Description: Law of excluded middle, in terms of restricted class abstractions.

Assertion

Ref	Expression
rabxm	|- A = ({x e. A | ph} u. {x e. A | -. ph})

Distinct variable group:   x,A

Proof of Theorem rabxm

Step	Hyp	Ref	Expression
1		rabid2 2254	. . 3 |- (A = {x e. A | (ph \/ -. ph)} <-> A.x e. A (ph \/ -. ph))
2		exmid 717	. . . 4 |- (ph \/ -. ph)
3	2	a1i 8	. . 3 |- (x e. A -> (ph \/ -. ph))
4	1, 3	mprgbir 2163	. 2 |- A = {x e. A | (ph \/ -. ph)}
5		unrab 2865	. 2 |- ({x e. A | ph} u. {x e. A | -. ph}) = {x e. A | (ph \/ -. ph)}
6	4, 5	eqtr4i 1911	1 |- A = ({x e. A | ph} u. {x e. A | -. ph})

Syntax hints:  -. wn 2   \/ wo 239   = wceq 1298   e. wcel 1300  {crab 2108   u. cun 2591

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294  df-un 2600

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