Theorem symdifeq2 4224

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

Assertion

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

Proof of Theorem symdifeq2

Step	Hyp	Ref	Expression
1		symdifeq1 4223	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶))
2		symdifcom 4222	. 2 ⊢ (𝐶 △ 𝐴) = (𝐴 △ 𝐶)
3		symdifcom 4222	. 2 ⊢ (𝐶 △ 𝐵) = (𝐵 △ 𝐶)
4	1, 2, 3	3eqtr4g 2883	1 ⊢ (𝐴 = 𝐵 → (𝐶 △ 𝐴) = (𝐶 △ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   △ 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-v 3498  df-dif 3941  df-un 3943  df-symdif 4221

This theorem is referenced by: (None)