Theorem nanbi2 1499

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof shortened by SF, 2-Jan-2018.)

Assertion

Ref	Expression
nanbi2	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓)))

Proof of Theorem nanbi2

Step	Hyp	Ref	Expression
1		nanbi1 1498	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))
2		nancom 1493	. 2 ⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜑 ⊼ 𝜒))
3		nancom 1493	. 2 ⊢ ((𝜒 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜒))
4	1, 2, 3	3bitr4g 314	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ⊼ wnan 1488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-nan 1489

This theorem is referenced by:  nanbi12  1500  nanbi2i  1502  nanbi2d  1505  nanass  1507