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Theorem 3anbi123d 1252
 Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
Hypotheses
Ref Expression
bi3d.1 (φ → (ψχ))
bi3d.2 (φ → (θτ))
bi3d.3 (φ → (ηζ))
Assertion
Ref Expression
3anbi123d (φ → ((ψ θ η) ↔ (χ τ ζ)))

Proof of Theorem 3anbi123d
StepHypRef Expression
1 bi3d.1 . . . 4 (φ → (ψχ))
2 bi3d.2 . . . 4 (φ → (θτ))
31, 2anbi12d 691 . . 3 (φ → ((ψ θ) ↔ (χ τ)))
4 bi3d.3 . . 3 (φ → (ηζ))
53, 4anbi12d 691 . 2 (φ → (((ψ θ) η) ↔ ((χ τ) ζ)))
6 df-3an 936 . 2 ((ψ θ η) ↔ ((ψ θ) η))
7 df-3an 936 . 2 ((χ τ ζ) ↔ ((χ τ) ζ))
85, 6, 73bitr4g 279 1 (φ → ((ψ θ η) ↔ (χ τ ζ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ 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:  3anbi12d  1253  3anbi13d  1254  3anbi23d  1255  ax11wdemo  1723  sbc3ang  3104  pw1equn  4331  pw1eqadj  4332  cenc  6181
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