Theorem nfsymdif 4105

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 4101	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		nfsymdif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfsymdif.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfdif 3988	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
5	3, 2	nfdif 3988	. . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴)
6	4, 5	nfun 4026	. 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7	1, 6	nfcxfr 2924	1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵)

Syntax hints:  Ⅎwnfc 2910   ∖ cdif 3822   ∪ cun 3823   △ csymdif 4100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-dif 3828  df-un 3830  df-symdif 4101

This theorem is referenced by: (None)