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Theorem xorbi12i 1625
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 328 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
43notbii 309 . 2 (¬ (𝜑𝜒) ↔ ¬ (𝜓𝜃))
5 df-xor 1613 . 2 ((𝜑𝜒) ↔ ¬ (𝜑𝜒))
6 df-xor 1613 . 2 ((𝜓𝜃) ↔ ¬ (𝜓𝜃))
74, 5, 63bitr4i 292 1 ((𝜑𝜒) ↔ (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wxo 1612
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 197  df-xor 1613
This theorem is referenced by:  hadcoma  1686  hadcomb  1687
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