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

Theorem xorcom 1295
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 657 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
2 ancom 257 . . . 4 ((𝜑𝜓) ↔ (𝜓𝜑))
32notbii 604 . . 3 (¬ (𝜑𝜓) ↔ ¬ (𝜓𝜑))
41, 3anbi12i 441 . 2 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ ((𝜓𝜑) ∧ ¬ (𝜓𝜑)))
5 df-xor 1283 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
6 df-xor 1283 . 2 ((𝜓𝜑) ↔ ((𝜓𝜑) ∧ ¬ (𝜓𝜑)))
74, 5, 63bitr4i 205 1 ((𝜑𝜓) ↔ (𝜓𝜑))
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  8712
  Copyright terms: Public domain W3C validator