Theorem nanbi2i 1535

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)

Hypothesis

Ref	Expression
nanbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
nanbi2i	⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))

Proof of Theorem nanbi2i

Step	Hyp	Ref	Expression
1		nanbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		nanbi2 1532	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ⊼ wnan 1521

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 1522

This theorem is used by:  nanass  1540  nabi2i  37001