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Theorem anxordi 1307
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 106 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜑)
2 df-xor 1283 . . . 4 (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∧ ¬ ((𝜑𝜓) ∧ (𝜑𝜒))))
32simplbi 263 . . 3 (((𝜑𝜓) ⊻ (𝜑𝜒)) → ((𝜑𝜓) ∨ (𝜑𝜒)))
4 simpl 106 . . . 4 ((𝜑𝜓) → 𝜑)
5 simpl 106 . . . 4 ((𝜑𝜒) → 𝜑)
64, 5jaoi 646 . . 3 (((𝜑𝜓) ∨ (𝜑𝜒)) → 𝜑)
73, 6syl 14 . 2 (((𝜑𝜓) ⊻ (𝜑𝜒)) → 𝜑)
8 ibar 289 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))
9 ibar 289 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
10 ibar 289 . . . 4 (𝜑 → (𝜒 ↔ (𝜑𝜒)))
119, 10xorbi12d 1289 . . 3 (𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
128, 11bitr3d 183 . 2 (𝜑 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
131, 7, 12pm5.21nii 630 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102  wo 639  wxo 1282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-xor 1283
This theorem is referenced by:  rpnegap  8713
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