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Metamath Proof Explorer

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 )