Theorem nfsymdif 4206

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 4202	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		nfsymdif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfsymdif.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfdif 4080	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
5	3, 2	nfdif 4080	. . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴)
6	4, 5	nfun 4120	. 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7	1, 6	nfcxfr 2922	1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵)

This proof depends on syntax axioms:  Ⅎwnfc 2909   ∖ cdif 3899   ∪ cun 3900   △ csymdif 4201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3455  df-dif 3905  df-un 3907  df-symdif 4202

This theorem is used by: (None)