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Theorem anxordi 1288
Description: Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
Assertion
Ref Expression
anxordi ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ)))

Proof of Theorem anxordi
StepHypRef Expression
1 simpl 102 . 2 ((φ (ψχ)) → φ)
2 df-xor 1266 . . . 4 (((φ ψ) ⊻ (φ χ)) ↔ (((φ ψ) (φ χ)) ¬ ((φ ψ) (φ χ))))
32simplbi 259 . . 3 (((φ ψ) ⊻ (φ χ)) → ((φ ψ) (φ χ)))
4 simpl 102 . . . 4 ((φ ψ) → φ)
5 simpl 102 . . . 4 ((φ χ) → φ)
64, 5jaoi 635 . . 3 (((φ ψ) (φ χ)) → φ)
73, 6syl 14 . 2 (((φ ψ) ⊻ (φ χ)) → φ)
8 ibar 285 . . 3 (φ → ((ψχ) ↔ (φ (ψχ))))
9 ibar 285 . . . 4 (φ → (ψ ↔ (φ ψ)))
10 ibar 285 . . . 4 (φ → (χ ↔ (φ χ)))
119, 10xorbi12d 1271 . . 3 (φ → ((ψχ) ↔ ((φ ψ) ⊻ (φ χ))))
128, 11bitr3d 179 . 2 (φ → ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ))))
131, 7, 12pm5.21nii 619 1 ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wo 628  wxo 1265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-xor 1266
This theorem is referenced by:  rpnegap  8350
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