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Theorem symdifeq1 3879
Description: Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem symdifeq1
StepHypRef Expression
1 difeq1 3754 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 difeq2 3755 . . 3 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2uneq12d 3801 . 2 (𝐴 = 𝐵 → ((𝐴𝐶) ∪ (𝐶𝐴)) = ((𝐵𝐶) ∪ (𝐶𝐵)))
4 df-symdif 3877 . 2 (𝐴𝐶) = ((𝐴𝐶) ∪ (𝐶𝐴))
5 df-symdif 3877 . 2 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
63, 4, 53eqtr4g 2710 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cdif 3604  cun 3605  csymdif 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-symdif 3877
This theorem is referenced by:  symdifeq2  3880
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