ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  an6 Structured version   GIF version

Theorem an6 1215
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 df-3an 886 . . . 4 ((φ ψ χ) ↔ ((φ ψ) χ))
2 df-3an 886 . . . 4 ((θ τ η) ↔ ((θ τ) η))
31, 2anbi12i 433 . . 3 (((φ ψ χ) (θ τ η)) ↔ (((φ ψ) χ) ((θ τ) η)))
4 an4 520 . . 3 ((((φ ψ) χ) ((θ τ) η)) ↔ (((φ ψ) (θ τ)) (χ η)))
5 an4 520 . . . 4 (((φ ψ) (θ τ)) ↔ ((φ θ) (ψ τ)))
65anbi1i 431 . . 3 ((((φ ψ) (θ τ)) (χ η)) ↔ (((φ θ) (ψ τ)) (χ η)))
73, 4, 63bitri 195 . 2 (((φ ψ χ) (θ τ η)) ↔ (((φ θ) (ψ τ)) (χ η)))
8 df-3an 886 . 2 (((φ θ) (ψ τ) (χ η)) ↔ (((φ θ) (ψ τ)) (χ η)))
97, 8bitr4i 176 1 (((φ ψ χ) (θ τ η)) ↔ ((φ θ) (ψ τ) (χ η)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-3an 886
This theorem is referenced by:  3an6  1216  elfzuzb  8634
  Copyright terms: Public domain W3C validator