Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1317 Structured version   Visualization version   GIF version

Theorem bnj1317 32372
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 2724 . 2 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2hbxfreq 2861 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wcel 2114  {cab 2716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811
This theorem is referenced by:  bnj1014  32512  bnj1145  32544  bnj1384  32583  bnj1398  32585  bnj1448  32598  bnj1450  32601  bnj1466  32604  bnj1463  32606  bnj1491  32608  bnj1497  32611  bnj1498  32612  bnj1520  32617  bnj1501  32618
  Copyright terms: Public domain W3C validator