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Theorem 3anbi123i 1104
Description: Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
Hypotheses
Ref Expression
bi3.1 (𝜑𝜓)
bi3.2 (𝜒𝜃)
bi3.3 (𝜏𝜂)
Assertion
Ref Expression
3anbi123i ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))

Proof of Theorem 3anbi123i
StepHypRef Expression
1 bi3.1 . . . 4 (𝜑𝜓)
2 bi3.2 . . . 4 (𝜒𝜃)
31, 2anbi12i 441 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
4 bi3.3 . . 3 (𝜏𝜂)
53, 4anbi12i 441 . 2 (((𝜑𝜒) ∧ 𝜏) ↔ ((𝜓𝜃) ∧ 𝜂))
6 df-3an 898 . 2 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∧ 𝜏))
7 df-3an 898 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∧ 𝜂))
85, 6, 73bitr4i 205 1 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by:  3anbi1i  1106  3anbi2i  1107  3anbi3i  1108  syl3anb  1189  ne3anior  2308
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