Theorem xorbi12d 1372

Description: Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)

Hypotheses

Ref	Expression
xorbi12d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
xorbi12d.2	⊢ (𝜑 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
xorbi12d	⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))

Proof of Theorem xorbi12d

Step	Hyp	Ref	Expression
1		xorbi12d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	xorbi1d 1371	. 2 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜃)))
3		xorbi12d.2	. . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
4	3	xorbi2d 1370	. 2 ⊢ (𝜑 → ((𝜒 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
5	2, 4	bitrd 187	1 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 104   ⊻ 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:  xorbi12i  1373  anxordi  1390  rpnegap  9622