Theorem 3anbi3d 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
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 1350	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1004

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 1006

This theorem is referenced by:  ceqsex3v  2846  ceqsex4v  2847  ceqsex8v  2849  vtocl3gaf  2873  mob  2988  ordsoexmid  4660  tfr1onlemaccex  6513  tfrcllemaccex  6526  fseq1m1p1  10329  pfxsuff1eqwrdeq  11279  summodc  11943  fsum3  11947  divalglemnn  12478  divalglemeunn  12481  divalglemex  12482  divalglemeuneg  12483  mhmlem  13700  ring1  14071  lmodlema  14305  ivthreinc  15368  dvmptfsum  15448