Theorem bnj1317 30892

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 2611	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	hbxfreq 2730	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  bnj1014  31030  bnj1145  31061  bnj1384  31100  bnj1398  31102  bnj1448  31115  bnj1450  31118  bnj1466  31121  bnj1463  31123  bnj1491  31125  bnj1497  31128  bnj1498  31129  bnj1520  31134  bnj1501  31135