Theorem 3anbi1d 1306

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
3anbi1d	⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))

Proof of Theorem 3anbi1d

Step	Hyp	Ref	Expression
1		3anbi1d.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		biidd 171	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃))
3	1, 2	3anbi12d 1303	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  vtocl3gaf  2795  ordsoexmid  4539  genpelxp  7452