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Theorem xorbi12d 1475
Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
xor12d.1 (𝜑 → (𝜓𝜒))
xor12d.2 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
xorbi12d (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Proof of Theorem xorbi12d
StepHypRef Expression
1 xor12d.1 . . . 4 (𝜑 → (𝜓𝜒))
2 xor12d.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2bibi12d 335 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
43notbid 308 . 2 (𝜑 → (¬ (𝜓𝜃) ↔ ¬ (𝜒𝜏)))
5 df-xor 1462 . 2 ((𝜓𝜃) ↔ ¬ (𝜓𝜃))
6 df-xor 1462 . 2 ((𝜒𝜏) ↔ ¬ (𝜒𝜏))
74, 5, 63bitr4g 303 1 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wxo 1461
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 1462
This theorem is referenced by:  hadbi123d  1531  cadbi123d  1546
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