Theorem nabi12i 37184

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

Hypotheses

Ref	Expression
nabi12i.1	⊢ (𝜑 ↔ 𝜓)
nabi12i.2	⊢ (𝜒 ↔ 𝜃)
nabi12i.3	⊢ (𝜓 ⊼ 𝜃)

Assertion

Ref	Expression
nabi12i	⊢ (𝜑 ⊼ 𝜒)

Proof of Theorem nabi12i

Step	Hyp	Ref	Expression
1		nabi12i.2	. 2 ⊢ (𝜒 ↔ 𝜃)
2		nabi12i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3		nabi12i.3	. . 3 ⊢ (𝜓 ⊼ 𝜃)
4	2, 3	nabi1i 37182	. 2 ⊢ (𝜑 ⊼ 𝜃)
5	1, 4	nabi2i 37183	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: (None)