Theorem xorbi12i 1314

Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Hypotheses

Ref	Expression
xorbi12.1	⊢ (φ ↔ ψ)
xorbi12.2	⊢ (χ ↔ θ)

Assertion

Ref	Expression
xorbi12i	⊢ ((φ ⊻ χ) ↔ (ψ ⊻ θ))

Proof of Theorem xorbi12i

Step	Hyp	Ref	Expression
1		xorbi12.1	. . . 4 ⊢ (φ ↔ ψ)
2		xorbi12.2	. . . 4 ⊢ (χ ↔ θ)
3	1, 2	bibi12i 306	. . 3 ⊢ ((φ ↔ χ) ↔ (ψ ↔ θ))
4	3	notbii 287	. 2 ⊢ (¬ (φ ↔ χ) ↔ ¬ (ψ ↔ θ))
5		df-xor 1305	. 2 ⊢ ((φ ⊻ χ) ↔ ¬ (φ ↔ χ))
6		df-xor 1305	. 2 ⊢ ((ψ ⊻ θ) ↔ ¬ (ψ ↔ θ))
7	4, 5, 6	3bitr4i 268	1 ⊢ ((φ ⊻ χ) ↔ (ψ ⊻ θ))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ⊻ wxo 1304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-xor 1305

This theorem is used by:  hadcoma  1388  hadcomb  1389  hadnot  1393