Theorem bnj1345

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

	Ref	Expression
Hypotheses	bnj1345.1	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
	bnj1345.2	⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
	bnj1345.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion	bnj1345	⊢ ( 𝜑 → ∃ 𝑥 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		bnj1345.1	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
2		bnj1345.2	⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
3		bnj1345.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4	1 3	bnj1275	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
5	4 2	bnj1198	⊢ ( 𝜑 → ∃ 𝑥 𝜃 )