Theorem xorbi2d 1380

Description: Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)

Hypothesis

Ref	Expression
xorbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
xorbi2d	⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒)))

Proof of Theorem xorbi2d

Step	Hyp	Ref	Expression
1		xorbid.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	orbi2d 790	. . 3 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒)))
3	1	anbi2d 464	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
4	3	notbid 667	. . 3 ⊢ (𝜑 → (¬ (𝜃 ∧ 𝜓) ↔ ¬ (𝜃 ∧ 𝜒)))
5	2, 4	anbi12d 473	. 2 ⊢ (𝜑 → (((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓)) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒))))
6		df-xor 1376	. 2 ⊢ ((𝜃 ⊻ 𝜓) ↔ ((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓)))
7		df-xor 1376	. 2 ⊢ ((𝜃 ⊻ 𝜒) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒)))
8	5, 6, 7	3bitr4g 223	1 ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ 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:  xorbi12d  1382