Theorem hbab 2749

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2402. See hbabg 2750 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)

Hypothesis

Ref	Expression
hbab.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

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

Proof of Theorem hbab

Step	Hyp	Ref	Expression
1		df-clab 2740	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbab.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsbw 2208	. 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 2099   ∈ wcel 2145  {cab 2739

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 1943  ax-6 2000  ax-7 2041  ax-11 2194

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740

This theorem is used by:  nfsab  2751  bnj1441  35470  bnj1309  35652