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Metamath Proof Explorer

Theorem nanbi1

Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by Wolf Lammen, 27-Jun-2020)

Ref Expression
Assertion nanbi1 ⊢ φ ↔ ψ → φ ⊼ χ ↔ ψ ⊼ χ

Proof

Step Hyp Ref Expression
1 imbi1 ⊢ φ ↔ ψ → φ → ¬ χ ↔ ψ → ¬ χ
2 dfnan2 ⊢ φ ⊼ χ ↔ φ → ¬ χ
3 dfnan2 ⊢ ψ ⊼ χ ↔ ψ → ¬ χ
4 1 2 3 3bitr4g ⊢ φ ↔ ψ → φ ⊼ χ ↔ ψ ⊼ χ