Theorem bnj1441 35237

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2403. See bnj1441g 35238 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1441.1	⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
bnj1441.2	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
bnj1441	⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

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

Proof of Theorem bnj1441

Step	Hyp	Ref	Expression
1		df-rab 3416	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		bnj1441.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
3		bnj1441.2	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
4	2, 3	hban 2334	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	hbab 2750	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6	1, 5	hbxfreq 2892	1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567   ∈ wcel 2142  {cab 2740  {crab 3415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416

This theorem is used by: (None)