Theorem hbabg 2752

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2404. See hbab 2751 for a version with more disjoint variable conditions, but not requiring ax-13 2404. (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 2742	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbabg.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsb 2556	. 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
4	1, 3	hbxfrbi 1858	1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2101   ∈ wcel 2147  {cab 2741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742

This theorem is used by:  nfsabg  2754  bnj1441g  35538