Theorem 3anbi3d 1355

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 1351	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))

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:  ceqsex3v  2859  ceqsex4v  2860  ceqsex8v  2862  vtocl3gaf  2886  mob  3002  ordsoexmid  4689  tfr1onlemaccex  6592  tfrcllemaccex  6605  fseq1m1p1  10451  pfxsuff1eqwrdeq  11416  summodc  12094  fsum3  12098  divalglemnn  12629  divalglemeunn  12632  divalglemex  12633  divalglemeuneg  12634  mhmlem  13867  ring1  14302  lmodlema  14566  ivthreinc  15636  dvmptfsum  15716