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Metamath Proof Explorer

Theorem bnj1276

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1276.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
bnj1276.2 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
bnj1276.3 ⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
bnj1276.4 ⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
Assertion bnj1276 ( 𝜃 → ∀ 𝑥 𝜃 )

Proof

Step Hyp Ref Expression
1 bnj1276.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2 bnj1276.2 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3 bnj1276.3 ⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
4 bnj1276.4 ⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
5 1 2 3 hb3an ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
6 4 5 hbxfrbi ⊢ ( 𝜃 → ∀ 𝑥 𝜃 )