Theorem nanbi2 1456

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed 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 1455	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))
2		nancom 1450	. 2 ⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜑 ⊼ 𝜒))
3		nancom 1450	. 2 ⊢ ((𝜒 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜒))
4	1, 2, 3	3bitr4g 303	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ⊼ wnan 1447

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 197  df-an 386  df-nan 1448

This theorem is referenced by:  nanbi12  1457  nanbi2i  1459  nanbi2d  1462  nabi2  32390  rp-fakenanass  37860