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Theorem an6 1261
 Description: Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
Assertion
Ref Expression
an6 (((φ ψ χ) (θ τ η)) ↔ ((φ θ) (ψ τ) (χ η)))

Proof of Theorem an6
StepHypRef Expression
1 an4 797 . . 3 ((((φ ψ) χ) ((θ τ) η)) ↔ (((φ ψ) (θ τ)) (χ η)))
2 an4 797 . . . 4 (((φ ψ) (θ τ)) ↔ ((φ θ) (ψ τ)))
32anbi1i 676 . . 3 ((((φ ψ) (θ τ)) (χ η)) ↔ (((φ θ) (ψ τ)) (χ η)))
41, 3bitri 240 . 2 ((((φ ψ) χ) ((θ τ) η)) ↔ (((φ θ) (ψ τ)) (χ η)))
5 df-3an 936 . . 3 ((φ ψ χ) ↔ ((φ ψ) χ))
6 df-3an 936 . . 3 ((θ τ η) ↔ ((θ τ) η))
75, 6anbi12i 678 . 2 (((φ ψ χ) (θ τ η)) ↔ (((φ ψ) χ) ((θ τ) η)))
8 df-3an 936 . 2 (((φ θ) (ψ τ) (χ η)) ↔ (((φ θ) (ψ τ)) (χ η)))
94, 7, 83bitr4i 268 1 (((φ ψ χ) (θ τ η)) ↔ ((φ θ) (ψ τ) (χ η)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936 This theorem is referenced by:  3an6  1262  fntxp  5804
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