Theorem 3anbi1i 1142

Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1i.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
3anbi1i	⊢ ((φ ∧ χ ∧ θ) ↔ (ψ ∧ χ ∧ θ))

Proof of Theorem 3anbi1i

Step	Hyp	Ref	Expression
1		3anbi1i.1	. 2 ⊢ (φ ↔ ψ)
2		biid 227	. 2 ⊢ (χ ↔ χ)
3		biid 227	. 2 ⊢ (θ ↔ θ)
4	1, 2, 3	3anbi123i 1140	1 ⊢ ((φ ∧ χ ∧ θ) ↔ (ψ ∧ χ ∧ θ))

Syntax hints:   ↔ wb 176   ∧ w3a 934

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-3an 936

This theorem is referenced by:  brsnsi1  5776