Theorem nfsymdif 4225

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 4221	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		nfsymdif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfsymdif.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfdif 4104	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
5	3, 2	nfdif 4104	. . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴)
6	4, 5	nfun 4143	. 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7	1, 6	nfcxfr 2977	1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵)

Syntax hints:  Ⅎwnfc 2963   ∖ cdif 3935   ∪ cun 3936   △ csymdif 4220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-dif 3941  df-un 3943  df-symdif 4221

This theorem is referenced by: (None)