Theorem hbabg 2810

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2389. See hbab 2809 for a version with more disjoint variable conditions, but not requiring ax-13 2389. (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 2799	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbabg.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsb 2566	. 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
4	1, 3	hbxfrbi 1824	1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1534  [wsb 2068   ∈ wcel 2113  {cab 2798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799

This theorem is referenced by:  nfsabg  2812  bnj1441g  32137