Theorem bnj1340

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

	Ref	Expression
Hypotheses	bnj1340.1	⊢ ( 𝜓 → ∃ 𝑥 𝜃 )
	bnj1340.2	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝜃 ) )
	bnj1340.3	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
Assertion	bnj1340	⊢ ( 𝜓 → ∃ 𝑥 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		bnj1340.1	⊢ ( 𝜓 → ∃ 𝑥 𝜃 )
2		bnj1340.2	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝜃 ) )
3		bnj1340.3	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
4	3 1	bnj596	⊢ ( 𝜓 → ∃ 𝑥 ( 𝜓 ∧ 𝜃 ) )
5	4 2	bnj1198	⊢ ( 𝜓 → ∃ 𝑥 𝜒 )