Theorem nanbi12 1532

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

Assertion

Ref	Expression
nanbi12	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)))

Proof of Theorem nanbi12

Step	Hyp	Ref	Expression
1		nanbi1 1530	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))
2		nanbi2 1531	. 2 ⊢ ((𝜒 ↔ 𝜃) → ((𝜓 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)))
3	1, 2	sylan9bb 518	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ⊼ wnan 1520

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-nan 1521

This theorem is used by:  nanbi12i  1535  nanbi12d  1538