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Theorem symdifeq1 4219
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 4090 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 difeq2 4091 . . 3 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2uneq12d 4138 . 2 (𝐴 = 𝐵 → ((𝐴𝐶) ∪ (𝐶𝐴)) = ((𝐵𝐶) ∪ (𝐶𝐵)))
4 df-symdif 4217 . 2 (𝐴𝐶) = ((𝐴𝐶) ∪ (𝐶𝐴))
5 df-symdif 4217 . 2 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
63, 4, 53eqtr4g 2879 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  cdif 3931  cun 3932  csymdif 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-symdif 4217
This theorem is referenced by:  symdifeq2  4220
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