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Theorem 3anbi123i 1140
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 ((φ χ τ) ↔ (ψ θ η))

Proof of Theorem 3anbi123i
StepHypRef Expression
1 bi3.1 . . . 4 (φψ)
2 bi3.2 . . . 4 (χθ)
31, 2anbi12i 678 . . 3 ((φ χ) ↔ (ψ θ))
4 bi3.3 . . 3 (τη)
53, 4anbi12i 678 . 2 (((φ χ) τ) ↔ ((ψ θ) η))
6 df-3an 936 . 2 ((φ χ τ) ↔ ((φ χ) τ))
7 df-3an 936 . 2 ((ψ θ η) ↔ ((ψ θ) η))
85, 6, 73bitr4i 268 1 ((φ χ τ) ↔ (ψ θ η))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358   w3a 934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
This theorem is referenced by:  3anbi1i  1142  3anbi2i  1143  3anbi3i  1144  syl3anb  1225  cadnot  1394  opksnelsik  4265  eloprabga  5578  restxp  5786  oqelins4  5794  xpassen  6057  mucass  6135  taddc  6229  letc  6231
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