Theorem hbabg 2727

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2372. See hbab 2726 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hbabg.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

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

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem hbabg

Step	Hyp	Ref	Expression
1		df-clab 2716	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbabg.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsb 2529	. 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
4	1, 3	hbxfrbi 1828	1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2068   ∈ wcel 2108  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716

This theorem is referenced by:  nfsabg  2729  bnj1441g  32721