Theorem 3anbi123d 1249

Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
bi3d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bi3d.2	⊢ (𝜑 → (𝜃 ↔ 𝜏))
bi3d.3	⊢ (𝜑 → (𝜂 ↔ 𝜁))

Assertion

Ref	Expression
3anbi123d	⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁)))

Proof of Theorem 3anbi123d

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		bi3d.2	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
3	1, 2	anbi12d 458	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏)))
4		bi3d.3	. . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁))
5	3, 4	anbi12d 458	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁)))
6		df-3an 927	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂))
7		df-3an 927	. 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁))
8	5, 6, 7	3bitr4g 222	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 925

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 927

This theorem is referenced by:  3anbi12d  1250  3anbi13d  1251  3anbi23d  1252  limeq  4213  smoeq  6069  tfrlemi1  6111  tfr1onlemaccex  6127  tfrcllemaccex  6140  ereq1  6313  updjud  6827  elinp  7094  iccshftr  9472  iccshftl  9474  iccdil  9476  icccntr  9478  fzaddel  9534  elfzomelpfzo  9703  seq3f1olemstep  9991  seq3f1olemp  9992  sumeq1  10805  isummolem2  10833  isummo  10834  zisum  10835  divalglemnn  11257  divalglemeunn  11260  divalglemeuneg  11262  dfgcd2  11342  isstruct2im  11565  isstruct2r  11566  fiinopn  11764