ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xorbi1d GIF version

Theorem xorbi1d 1359
Description: Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
Hypothesis
Ref Expression
xorbid.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
xorbi1d (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Proof of Theorem xorbi1d
StepHypRef Expression
1 xorbid.1 . . . 4 (𝜑 → (𝜓𝜒))
21orbi1d 780 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
31anbi1d 460 . . . 4 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
43notbid 656 . . 3 (𝜑 → (¬ (𝜓𝜃) ↔ ¬ (𝜒𝜃)))
52, 4anbi12d 464 . 2 (𝜑 → (((𝜓𝜃) ∧ ¬ (𝜓𝜃)) ↔ ((𝜒𝜃) ∧ ¬ (𝜒𝜃))))
6 df-xor 1354 . 2 ((𝜓𝜃) ↔ ((𝜓𝜃) ∧ ¬ (𝜓𝜃)))
7 df-xor 1354 . 2 ((𝜒𝜃) ↔ ((𝜒𝜃) ∧ ¬ (𝜒𝜃)))
85, 6, 73bitr4g 222 1 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  wxo 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-xor 1354
This theorem is referenced by:  xorbi12d  1360
  Copyright terms: Public domain W3C validator