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Theorem xoranor 1251
Description: One way of defining exclusive or. Equivalent to df-xor 1250. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
Assertion
Ref Expression
xoranor ((φψ) ↔ ((φ ψ) φ ¬ ψ)))

Proof of Theorem xoranor
StepHypRef Expression
1 df-xor 1250 . . 3 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
2 ax-ia3 101 . . . . . . 7 (φ → (ψ → (φ ψ)))
32con3d 548 . . . . . 6 (φ → (¬ (φ ψ) → ¬ ψ))
4 olc 619 . . . . . 6 ψ → (¬ φ ¬ ψ))
53, 4syl6 29 . . . . 5 (φ → (¬ (φ ψ) → (¬ φ ¬ ψ)))
6 pm3.21 251 . . . . . . 7 (ψ → (φ → (φ ψ)))
76con3d 548 . . . . . 6 (ψ → (¬ (φ ψ) → ¬ φ))
8 orc 620 . . . . . 6 φ → (¬ φ ¬ ψ))
97, 8syl6 29 . . . . 5 (ψ → (¬ (φ ψ) → (¬ φ ¬ ψ)))
105, 9jaoi 623 . . . 4 ((φ ψ) → (¬ (φ ψ) → (¬ φ ¬ ψ)))
1110imdistani 422 . . 3 (((φ ψ) ¬ (φ ψ)) → ((φ ψ) φ ¬ ψ)))
121, 11sylbi 114 . 2 ((φψ) → ((φ ψ) φ ¬ ψ)))
13 pm3.14 657 . . . 4 ((¬ φ ¬ ψ) → ¬ (φ ψ))
1413anim2i 324 . . 3 (((φ ψ) φ ¬ ψ)) → ((φ ψ) ¬ (φ ψ)))
1514, 1sylibr 137 . 2 (((φ ψ) φ ¬ ψ)) → (φψ))
1612, 15impbii 117 1 ((φψ) ↔ ((φ ψ) φ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616  wxo 1249
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 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-xor 1250
This theorem is referenced by:  excxor  1252
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