Theorem bnj1101

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

	Ref	Expression
Hypotheses	bnj1101.1	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
	bnj1101.2	⊢ ( 𝜒 → 𝜑 )
Assertion	bnj1101	⊢ ∃ 𝑥 ( 𝜒 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj1101.1	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
2		bnj1101.2	⊢ ( 𝜒 → 𝜑 )
3		pm3.42	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ∧ 𝜑 ) → 𝜓 ) )
4	1 3	bnj101	⊢ ∃ 𝑥 ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5	2	pm4.71i	⊢ ( 𝜒 ↔ ( 𝜒 ∧ 𝜑 ) )
6	5	imbi1i	⊢ ( ( 𝜒 → 𝜓 ) ↔ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 ) )
7	6	exbii	⊢ ( ∃ 𝑥 ( 𝜒 → 𝜓 ) ↔ ∃ 𝑥 ( ( 𝜒 ∧ 𝜑 ) → 𝜓 ) )
8	4 7	mpbir	⊢ ∃ 𝑥 ( 𝜒 → 𝜓 )