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Theorem bnj1317 32081
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 2805 . 2 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2hbxfreq 2940 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528   = wceq 1530  wcel 2107  {cab 2797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891
This theorem is referenced by:  bnj1014  32221  bnj1145  32253  bnj1384  32292  bnj1398  32294  bnj1448  32307  bnj1450  32310  bnj1466  32313  bnj1463  32315  bnj1491  32317  bnj1497  32320  bnj1498  32321  bnj1520  32326  bnj1501  32327
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