Theorem nfsymdif 3234

Description: Hypothesis builder for symmetric difference. (Contributed by SF, 2-Jan-2018.)

Hypotheses

Ref	Expression
nfbool.1	⊢ ℲxA
nfbool.2	⊢ ℲxB

Assertion

Ref	Expression
nfsymdif	⊢ Ⅎx(A ⊕ B)

Proof of Theorem nfsymdif

Step	Hyp	Ref	Expression
1		df-symdif 3217	. 2 ⊢ (A ⊕ B) = ((A ∖ B) ∪ (B ∖ A))
2		nfbool.1	. . . 4 ⊢ ℲxA
3		nfbool.2	. . . 4 ⊢ ℲxB
4	2, 3	nfdif 3233	. . 3 ⊢ Ⅎx(A ∖ B)
5	3, 2	nfdif 3233	. . 3 ⊢ Ⅎx(B ∖ A)
6	4, 5	nfun 3232	. 2 ⊢ Ⅎx((A ∖ B) ∪ (B ∖ A))
7	1, 6	nfcxfr 2487	1 ⊢ Ⅎx(A ⊕ B)

Syntax hints:  Ⅎwnfc 2477   ∖ cdif 3207   ∪ cun 3208   ⊕ csymdif 3210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217

This theorem is referenced by: (None)