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Theorem xorbi12i 1647
Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
xorbi12.1 (𝜑𝜓)
xorbi12.2 (𝜒𝜃)
Assertion
Ref Expression
xorbi12i ((𝜑𝜒) ↔ (𝜓𝜃))

Proof of Theorem xorbi12i
StepHypRef Expression
1 xorbi12.1 . . . 4 (𝜑𝜓)
2 xorbi12.2 . . . 4 (𝜒𝜃)
31, 2bibi12i 331 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
43notbii 312 . 2 (¬ (𝜑𝜒) ↔ ¬ (𝜓𝜃))
5 df-xor 1635 . 2 ((𝜑𝜒) ↔ ¬ (𝜑𝜒))
6 df-xor 1635 . 2 ((𝜓𝜃) ↔ ¬ (𝜓𝜃))
74, 5, 63bitr4i 295 1 ((𝜑𝜒) ↔ (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wxo 1634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-xor 1635
This theorem is referenced by:  hadcoma  1709  hadcomb  1710  symdifass  4050
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