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Metamath Proof Explorer

Theorem xorbi12d

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

Ref Expression
Hypotheses xor12d.1 ⊢ φ → ψ ↔ χ
xor12d.2 ⊢ φ → θ ↔ τ
Assertion xorbi12d ⊢ φ → ψ ⊻ θ ↔ χ ⊻ τ

Proof

Step Hyp Ref Expression
1 xor12d.1 ⊢ φ → ψ ↔ χ
2 xor12d.2 ⊢ φ → θ ↔ τ
3 1 2 bibi12d ⊢ φ → ψ ↔ θ ↔ χ ↔ τ
4 3 notbid ⊢ φ → ¬ ψ ↔ θ ↔ ¬ χ ↔ τ
5 df-xor ⊢ ψ ⊻ θ ↔ ¬ ψ ↔ θ
6 df-xor ⊢ χ ⊻ τ ↔ ¬ χ ↔ τ
7 4 5 6 3bitr4g ⊢ φ → ψ ⊻ θ ↔ χ ⊻ τ