Theorem bnj1317 34814

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 2721	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	hbxfreq 2870	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537   ∈ wcel 2106  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814

This theorem is referenced by:  bnj1014  34954  bnj1145  34986  bnj1384  35025  bnj1398  35027  bnj1448  35040  bnj1450  35043  bnj1466  35046  bnj1463  35048  bnj1491  35050  bnj1497  35053  bnj1498  35054  bnj1520  35059  bnj1501  35060