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Theorem bnj1317 31409
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 = {𝑥𝜑}
Assertion
Ref Expression
bnj1317 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem bnj1317
StepHypRef Expression
1 bnj1317.1 . 2 𝐴 = {𝑥𝜑}
2 hbab1 2788 . 2 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2hbxfreq 2907 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651   = wceq 1653  wcel 2157  {cab 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795
This theorem is referenced by:  bnj1014  31547  bnj1145  31578  bnj1384  31617  bnj1398  31619  bnj1448  31632  bnj1450  31635  bnj1466  31638  bnj1463  31640  bnj1491  31642  bnj1497  31645  bnj1498  31646  bnj1520  31651  bnj1501  31652
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