Theorem nfsymdif 3848

Description: Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfsymdif	⊢ Ⅎ𝑥(𝐴 △ 𝐵)

Proof of Theorem nfsymdif

Step	Hyp	Ref	Expression
1		df-symdif 3844	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		nfsymdif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfsymdif.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfdif 3731	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
5	3, 2	nfdif 3731	. . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴)
6	4, 5	nfun 3769	. 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7	1, 6	nfcxfr 2762	1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵)

Syntax hints:  Ⅎwnfc 2751   ∖ cdif 3571   ∪ cun 3572   △ csymdif 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-dif 3577  df-un 3579  df-symdif 3844

This theorem is referenced by: (None)