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 )