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Theorem xorbi12i 1515
Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
Hypotheses
Ref Expression
xorbi12.1 (𝜑𝜓)
xorbi12.2 (𝜒𝜃)
Assertion
Ref Expression
xorbi12i ((𝜑𝜒) ↔ (𝜓𝜃))

Proof of Theorem xorbi12i
StepHypRef Expression
1 df-xor 1502 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑𝜒))
2 xorbi12.1 . . . 4 (𝜑𝜓)
3 xorbi12.2 . . . 4 (𝜒𝜃)
42, 3bibi12i 342 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
51, 4xchbinx 336 . 2 ((𝜑𝜒) ↔ ¬ (𝜓𝜃))
6 df-xor 1502 . 2 ((𝜓𝜃) ↔ ¬ (𝜓𝜃))
75, 6bitr4i 280 1 ((𝜑𝜒) ↔ (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wxo 1501
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 209  df-xor 1502
This theorem is referenced by:  hadcomaOLD  1600  hadcomb  1601  symdifass  4206
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