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Theorem bnj1317 29952
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 2598 . 2 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2hbxfreq 2716 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472   = wceq 1474  wcel 1976  {cab 2595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605
This theorem is referenced by:  bnj1014  30090  bnj1145  30121  bnj1384  30160  bnj1398  30162  bnj1448  30175  bnj1450  30178  bnj1466  30181  bnj1463  30183  bnj1491  30185  bnj1497  30188  bnj1498  30189  bnj1520  30194  bnj1501  30195
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