Theorem hbab1 2342

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
hbab1	⊢ (y ∈ {x ∣ φ} → ∀x y ∈ {x ∣ φ})

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem hbab1

Step	Hyp	Ref	Expression
1		df-clab 2340	. 2 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
2		hbs1 2105	. 2 ⊢ ([y / x]φ → ∀x[y / x]φ)
3	1, 2	hbxfrbi 1568	1 ⊢ (y ∈ {x ∣ φ} → ∀x y ∈ {x ∣ φ})

Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648   ∈ wcel 1710  {cab 2339

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  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340

This theorem is referenced by:  nfsab1  2343  abeq2  2458