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Metamath Proof Explorer

Theorem nanbi12i

Description: Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018)

Ref Expression
Hypotheses nanbii.1 ⊢ φ ↔ ψ
nanbi12i.2 ⊢ χ ↔ θ
Assertion nanbi12i ⊢ φ ⊼ χ ↔ ψ ⊼ θ

Proof

Step Hyp Ref Expression
1 nanbii.1 ⊢ φ ↔ ψ
2 nanbi12i.2 ⊢ χ ↔ θ
3 nanbi12 ⊢ φ ↔ ψ ∧ χ ↔ θ → φ ⊼ χ ↔ ψ ⊼ θ
4 1 2 3 mp2an ⊢ φ ⊼ χ ↔ ψ ⊼ θ