Theorem hbab 2344

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)

Hypothesis

Ref	Expression
hbab.1	⊢ (φ → ∀xφ)

Assertion

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

Distinct variable group:   x,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbab

Step	Hyp	Ref	Expression
1		df-clab 2340	. 2 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ)
2		hbab.1	. . 3 ⊢ (φ → ∀xφ)
3	2	hbsb 2110	. 2 ⊢ ([z / y]φ → ∀x[z / y]φ)
4	1, 3	hbxfrbi 1568	1 ⊢ (z ∈ {y ∣ φ} → ∀x z ∈ {y ∣ 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:  nfsab  2345  hboprab1  5559  hboprab2  5560  hboprab3  5561  hboprab  5562