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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 ) )