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  2844  ceqsex4v  2845  ceqsex8v  2847  vtocl3gaf  2871  mob  2986  ordsoexmid  4658  tfr1onlemaccex  6509  tfrcllemaccex  6522  fseq1m1p1  10320  pfxsuff1eqwrdeq  11270  summodc  11934  fsum3  11938  divalglemnn  12469  divalglemeunn  12472  divalglemex  12473  divalglemeuneg  12474  mhmlem  13691  ring1  14062  lmodlema  14296  ivthreinc  15359  dvmptfsum  15439