Theorem 3anbi123d 1190

Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
bi3d.1	⊢ (φ → (ψ ↔ χ))
bi3d.2	⊢ (φ → (θ ↔ τ))
bi3d.3	⊢ (φ → (η ↔ ζ))

Assertion

Ref	Expression
3anbi123d	⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ ζ)))

Proof of Theorem 3anbi123d

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 ⊢ (φ → (ψ ↔ χ))
2		bi3d.2	. . . 4 ⊢ (φ → (θ ↔ τ))
3	1, 2	anbi12d 445	. . 3 ⊢ (φ → ((ψ ∧ θ) ↔ (χ ∧ τ)))
4		bi3d.3	. . 3 ⊢ (φ → (η ↔ ζ))
5	3, 4	anbi12d 445	. 2 ⊢ (φ → (((ψ ∧ θ) ∧ η) ↔ ((χ ∧ τ) ∧ ζ)))
6		df-3an 873	. 2 ⊢ ((ψ ∧ θ ∧ η) ↔ ((ψ ∧ θ) ∧ η))
7		df-3an 873	. 2 ⊢ ((χ ∧ τ ∧ ζ) ↔ ((χ ∧ τ) ∧ ζ))
8	5, 6, 7	3bitr4g 212	1 ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ ζ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871

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 873

This theorem is referenced by:  3anbi12d  1191  3anbi13d  1192  3anbi23d  1193  sbc3ang  2793  limeq  4059  smoeq  5823  tfrlemi1  5863  ereq1  6020  elinp  6322