Theorem 3anbi3d 1352

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

Hypothesis

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

Assertion

Ref	Expression
3anbi3d	⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))

Proof of Theorem 3anbi3d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃))
2		3anbi1d.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	3anbi13d 1348	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1002

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 1004

This theorem is referenced by:  ceqsex3v  2843  ceqsex4v  2844  ceqsex8v  2846  vtocl3gaf  2870  mob  2985  ordsoexmid  4654  tfr1onlemaccex  6500  tfrcllemaccex  6513  fseq1m1p1  10303  pfxsuff1eqwrdeq  11246  summodc  11909  fsum3  11913  divalglemnn  12444  divalglemeunn  12447  divalglemex  12448  divalglemeuneg  12449  mhmlem  13666  ring1  14037  lmodlema  14271  ivthreinc  15334  dvmptfsum  15414