Theorem hbe1w 1708

Description: Weak version of hbe1 1731. See comments for ax6w 1717. 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 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
2		hbn1w.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	notbid 285	. . 3 ⊢ (x = y → (¬ φ ↔ ¬ ψ))
4	3	hbn1w 1706	. 2 ⊢ (¬ ∀x ¬ φ → ∀x ¬ ∀x ¬ φ)
5	1, 4	hbxfrbi 1568	1 ⊢ (∃xφ → ∀x∃xφ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)