Theorem elsymdifxor 3850

Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.)

Assertion

Ref	Expression
elsymdifxor	⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Proof of Theorem elsymdifxor

Step	Hyp	Ref	Expression
1		xnor 1466	. . 3 ⊢ ((𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
2	1	notbii 310	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ¬ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
3		elsymdif 3849	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
4		notnotb 304	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶) ↔ ¬ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
5	2, 3, 4	3bitr4i 292	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 196   ⊻ wxo 1464   ∈ wcel 1990   △ 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-xor 1465  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-v 3202  df-dif 3577  df-un 3579  df-symdif 3844

This theorem is referenced by:  dfsymdif2  3851