Theorem nabi1i 37182

Description: Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)

Hypotheses

Ref	Expression
nabi1i.1	⊢ (𝜑 ↔ 𝜓)
nabi1i.2	⊢ (𝜓 ⊼ 𝜒)

Assertion

Ref	Expression
nabi1i	⊢ (𝜑 ⊼ 𝜒)

Proof of Theorem nabi1i

Step	Hyp	Ref	Expression
1		nabi1i.2	. 2 ⊢ (𝜓 ⊼ 𝜒)
2		nabi1i.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	bicomi 227	. . 3 ⊢ (𝜓 ↔ 𝜑)
4	3	nanbi1i 1533	. 2 ⊢ ((𝜓 ⊼ 𝜒) ↔ (𝜑 ⊼ 𝜒))
5	1, 4	mpbi 233	1 ⊢ (𝜑 ⊼ 𝜒)

This proof depends on syntax axioms:   ↔ wb 209   ⊼ 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 402  df-nan 1521

This theorem is used by:  nabi12i  37184