Theorem hbe1w

Description: Weak version of hbe1 . See comments for ax10w . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypothesis	hbn1w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	hbe1w	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbn1w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
3	1	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4	3	hbn1w	⊢ ( ¬ ∀ 𝑥 ¬ 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ 𝜑 )
5	2 4	hbxfrbi	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )