Theorem nanbi12i 1536

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

Hypotheses

Ref	Expression
nanbii.1	⊢ (𝜑 ↔ 𝜓)
nanbi12i.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
nanbi12i	⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))

Proof of Theorem nanbi12i

Step	Hyp	Ref	Expression
1		nanbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		nanbi12i.2	. 2 ⊢ (𝜒 ↔ 𝜃)
3		nanbi12 1533	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)))
4	1, 2, 3	mp2an 705	1 ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))

This proof depends on syntax axioms:   ↔ wb 209   ⊼ wnan 1521

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 402  df-nan 1522

This theorem is used by: (None)