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Metamath Proof Explorer

Theorem symdifeq2

Description: Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012)

Ref Expression
Assertion symdifeq2 ( 𝐴 = 𝐵 → ( 𝐶 △ 𝐴 ) = ( 𝐶 △ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 symdifeq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 △ 𝐶 ) = ( 𝐵 △ 𝐶 ) )
2 symdifcom ⊢ ( 𝐶 △ 𝐴 ) = ( 𝐴 △ 𝐶 )
3 symdifcom ⊢ ( 𝐶 △ 𝐵 ) = ( 𝐵 △ 𝐶 )
4 1 2 3 3eqtr4g ⊢ ( 𝐴 = 𝐵 → ( 𝐶 △ 𝐴 ) = ( 𝐶 △ 𝐵 ) )