Theorem bnj1317 35218

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1317.1	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
bnj1317	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bnj1317

Step	Hyp	Ref	Expression
1		bnj1317.1	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
2		hbab1 2749	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	hbxfreq 2892	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2142  {cab 2740

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-12 2212  ax-ext 2734

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

This theorem is used by:  bnj1014  35358  bnj1145  35390  bnj1384  35429  bnj1398  35431  bnj1448  35444  bnj1450  35447  bnj1466  35450  bnj1463  35452  bnj1491  35454  bnj1497  35457  bnj1498  35458  bnj1520  35463  bnj1501  35464