Theorem bnj1275

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

	Ref	Expression
Hypotheses	bnj1275.1	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
	bnj1275.2	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion	bnj1275	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1275.1	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
2		bnj1275.2	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2 1	bnj596	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
4		3anass	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
5	3 4	bnj1198	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )