Theorem hbe1w 2047

Description: Weak version of hbe1 2142. See comments for ax10w 2128. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
hbe1w	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem hbe1w

Step	Hyp	Ref	Expression
1		df-ex 1779	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1w.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	notbid 318	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	hbn1w 2045	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
5	1, 4	hbxfrbi 1824	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)