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

Theorem xorcom 1262
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 634 . . 3 ((φ ψ) ↔ (ψ φ))
2 ancom 253 . . . 4 ((φ ψ) ↔ (ψ φ))
32notbii 581 . . 3 (¬ (φ ψ) ↔ ¬ (ψ φ))
41, 3anbi12i 436 . 2 (((φ ψ) ¬ (φ ψ)) ↔ ((ψ φ) ¬ (ψ φ)))
5 df-xor 1252 . 2 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
6 df-xor 1252 . 2 ((ψφ) ↔ ((ψ φ) ¬ (ψ φ)))
74, 5, 63bitr4i 201 1 ((φψ) ↔ (ψφ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wo 616  wxo 1251
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 1252
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator