Theorem bnj1317 34956

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 2722	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	hbxfreq 2865	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810

This theorem is referenced by:  bnj1014  35096  bnj1145  35128  bnj1384  35167  bnj1398  35169  bnj1448  35182  bnj1450  35185  bnj1466  35188  bnj1463  35190  bnj1491  35192  bnj1497  35195  bnj1498  35196  bnj1520  35201  bnj1501  35202