Theorem nanbi2 1296

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 1295	. 2 ⊢ ((φ ↔ ψ) → ((φ ⊼ χ) ↔ (ψ ⊼ χ)))
2		nancom 1290	. 2 ⊢ ((χ ⊼ φ) ↔ (φ ⊼ χ))
3		nancom 1290	. 2 ⊢ ((χ ⊼ ψ) ↔ (ψ ⊼ χ))
4	1, 2, 3	3bitr4g 279	1 ⊢ ((φ ↔ ψ) → ((χ ⊼ φ) ↔ (χ ⊼ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by:  nanbi12  1297  nanbi2i  1299  nanbi2d  1302