Theorem nanbi2d 1302

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

Hypothesis

Ref	Expression
nanbid.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
nanbi2d	⊢ (φ → ((θ ⊼ ψ) ↔ (θ ⊼ χ)))

Proof of Theorem nanbi2d

Step	Hyp	Ref	Expression
1		nanbid.1	. 2 ⊢ (φ → (ψ ↔ χ))
2		nanbi2 1296	. 2 ⊢ ((ψ ↔ χ) → ((θ ⊼ ψ) ↔ (θ ⊼ χ)))
3	1, 2	syl 15	1 ⊢ (φ → ((θ ⊼ ψ) ↔ (θ ⊼ χ)))

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: (None)