Theorem 3anbi2d 1354

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

Hypothesis

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

Assertion

Ref	Expression
3anbi2d	⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏)))

Proof of Theorem 3anbi2d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃))
2		3anbi1d.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	3anbi12d 1350	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  vtocl3gaf  2874  ordsoexmid  4666  ereq2  6753  genpelxp  7774  seq3f1olemp  10823  qexpclz  10868  mhmlem  13764  opprsubgg  14161  lmodlema  14371  ivthreinc  15439  incistruhgr  16014  issubgr2  16182