Theorem bnj596

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

	Ref	Expression
Hypotheses	bnj596.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bnj596.2	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
Assertion	bnj596	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj596.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bnj596.2	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
3	2	ancli	⊢ ( 𝜑 → ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )
4	1	nf5i	⊢ Ⅎ 𝑥 𝜑
5	4	19.42	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )
6	3 5	sylibr	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )