Theorem nfsymdif 4222

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 4218	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		nfsymdif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfsymdif.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfdif 4101	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
5	3, 2	nfdif 4101	. . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴)
6	4, 5	nfun 4140	. 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7	1, 6	nfcxfr 2975	1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵)

Syntax hints:  Ⅎwnfc 2961   ∖ cdif 3932   ∪ cun 3933   △ csymdif 4217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-dif 3938  df-un 3940  df-symdif 4218

This theorem is referenced by: (None)