Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xorbi12d GIF version

Theorem xorbi12d 1289
 Description: Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
Hypotheses
Ref Expression
xorbi12d.1 (𝜑 → (𝜓𝜒))
xorbi12d.2 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
xorbi12d (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Proof of Theorem xorbi12d
StepHypRef Expression
1 xorbi12d.1 . . 3 (𝜑 → (𝜓𝜒))
21xorbi1d 1288 . 2 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
3 xorbi12d.2 . . 3 (𝜑 → (𝜃𝜏))
43xorbi2d 1287 . 2 (𝜑 → ((𝜒𝜃) ↔ (𝜒𝜏)))
52, 4bitrd 181 1 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ⊻ 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:  xorbi12i  1290  anxordi  1307  rpnegap  8712
 Copyright terms: Public domain W3C validator