ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xorcom GIF version

Theorem xorcom 1388
Description: is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
Assertion
Ref Expression
xorcom ((𝜑𝜓) ↔ (𝜓𝜑))

Proof of Theorem xorcom
StepHypRef Expression
1 orcom 728 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
2 ancom 266 . . . 4 ((𝜑𝜓) ↔ (𝜓𝜑))
32notbii 668 . . 3 (¬ (𝜑𝜓) ↔ ¬ (𝜓𝜑))
41, 3anbi12i 460 . 2 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ ((𝜓𝜑) ∧ ¬ (𝜓𝜑)))
5 df-xor 1376 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
6 df-xor 1376 . 2 ((𝜓𝜑) ↔ ((𝜓𝜑) ∧ ¬ (𝜓𝜑)))
74, 5, 63bitr4i 212 1 ((𝜑𝜓) ↔ (𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 708  wxo 1375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-xor 1376
This theorem is referenced by:  rpnegap  9682
  Copyright terms: Public domain W3C validator