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Metamath Proof Explorer

Theorem anbi12d

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993)

Ref Expression
Hypotheses anbi12d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
anbi12d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion anbi12d ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 anbi12d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 anbi12d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3 1 anbi1d ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )
4 2 anbi2d ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜏 ) ) )
5 3 4 bitrd ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜏 ) ) )