Theorem bnj1096

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

	Ref	Expression
Hypotheses	bnj1096.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bnj1096.2	⊢ ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) )
Assertion	bnj1096	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj1096.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bnj1096.2	⊢ ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) )
3		ax-5	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
4		ax-5	⊢ ( 𝜃 → ∀ 𝑥 𝜃 )
5		ax-5	⊢ ( 𝜏 → ∀ 𝑥 𝜏 )
6	3 4 5 1	bnj982	⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) → ∀ 𝑥 ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) )
7	2 6	hbxfrbi	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )