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Theorem anxordi 1378
Description: Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
Assertion
Ref Expression
anxordi ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))

Proof of Theorem anxordi
StepHypRef Expression
1 simpl 108 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜑)
2 df-xor 1354 . . . 4 (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∧ ¬ ((𝜑𝜓) ∧ (𝜑𝜒))))
32simplbi 272 . . 3 (((𝜑𝜓) ⊻ (𝜑𝜒)) → ((𝜑𝜓) ∨ (𝜑𝜒)))
4 simpl 108 . . . 4 ((𝜑𝜓) → 𝜑)
5 simpl 108 . . . 4 ((𝜑𝜒) → 𝜑)
64, 5jaoi 705 . . 3 (((𝜑𝜓) ∨ (𝜑𝜒)) → 𝜑)
73, 6syl 14 . 2 (((𝜑𝜓) ⊻ (𝜑𝜒)) → 𝜑)
8 ibar 299 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))
9 ibar 299 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
10 ibar 299 . . . 4 (𝜑 → (𝜒 ↔ (𝜑𝜒)))
119, 10xorbi12d 1360 . . 3 (𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
128, 11bitr3d 189 . 2 (𝜑 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
131, 7, 12pm5.21nii 693 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 697  wxo 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-xor 1354
This theorem is referenced by:  rpnegap  9486
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