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Metamath Proof Explorer

Theorem 19.26-3an

Description: Theorem 19.26 with triple conjunction. (Contributed by NM, 13-Sep-2011)

Ref Expression
Assertion 19.26-3an ⊢ ∀ x φ ∧ ψ ∧ χ ↔ ∀ x φ ∧ ∀ x ψ ∧ ∀ x χ

Proof

Step Hyp Ref Expression
1 19.26 ⊢ ∀ x φ ∧ ψ ↔ ∀ x φ ∧ ∀ x ψ
2 1 anbi1i ⊢ ∀ x φ ∧ ψ ∧ ∀ x χ ↔ ∀ x φ ∧ ∀ x ψ ∧ ∀ x χ
3 df-3an ⊢ φ ∧ ψ ∧ χ ↔ φ ∧ ψ ∧ χ
4 3 albii ⊢ ∀ x φ ∧ ψ ∧ χ ↔ ∀ x φ ∧ ψ ∧ χ
5 19.26 ⊢ ∀ x φ ∧ ψ ∧ χ ↔ ∀ x φ ∧ ψ ∧ ∀ x χ
6 4 5 bitri ⊢ ∀ x φ ∧ ψ ∧ χ ↔ ∀ x φ ∧ ψ ∧ ∀ x χ
7 df-3an ⊢ ∀ x φ ∧ ∀ x ψ ∧ ∀ x χ ↔ ∀ x φ ∧ ∀ x ψ ∧ ∀ x χ
8 2 6 7 3bitr4i ⊢ ∀ x φ ∧ ψ ∧ χ ↔ ∀ x φ ∧ ∀ x ψ ∧ ∀ x χ