Theorem bnj1521

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

	Ref	Expression
Hypotheses	bnj1521.1	⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
	bnj1521.2	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
	bnj1521.3	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	bnj1521	⊢ ( 𝜒 → ∃ 𝑥 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		bnj1521.1	⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
2		bnj1521.2	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
3		bnj1521.3	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
4	1	bnj1196	⊢ ( 𝜒 → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
5	4 2 3	bnj1345	⊢ ( 𝜒 → ∃ 𝑥 𝜃 )