Theorem hbabg 2730

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. See hbab 2729 for a version with more disjoint variable conditions, but not requiring ax-13 2382. (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 2720	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbabg.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsb 2534	. 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
4	1, 3	hbxfrbi 1833	1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1546  [wsb 2074   ∈ wcel 2121  {cab 2719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-11 2170  ax-12 2191  ax-13 2382

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720

This theorem is referenced by:  nfsabg  2732  bnj1441g  35038