Theorem xorbi2d 1370

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 780	. . 3 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒)))
3	1	anbi2d 460	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
4	3	notbid 657	. . 3 ⊢ (𝜑 → (¬ (𝜃 ∧ 𝜓) ↔ ¬ (𝜃 ∧ 𝜒)))
5	2, 4	anbi12d 465	. 2 ⊢ (𝜑 → (((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓)) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒))))
6		df-xor 1366	. 2 ⊢ ((𝜃 ⊻ 𝜓) ↔ ((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓)))
7		df-xor 1366	. 2 ⊢ ((𝜃 ⊻ 𝜒) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒)))
8	5, 6, 7	3bitr4g 222	1 ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ⊻ wxo 1365

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by:  xorbi12d  1372