Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralab Structured version   Visualization version   GIF version

Theorem ralab 3365
 Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
ralab (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
Distinct variable groups:   𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑥,𝑦)

Proof of Theorem ralab
StepHypRef Expression
1 df-ral 2916 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒))
2 vex 3201 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 3348 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54imbi1i 339 . . 3 ((𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ (𝜓𝜒))
65albii 1746 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ ∀𝑥(𝜓𝜒))
71, 6bitri 264 1 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1480   ∈ wcel 1989  {cab 2607  ∀wral 2911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-v 3200 This theorem is referenced by:  ralrnmpt2  6772  funcnvuni  7116  kardex  8754  karden  8755  fimaxre3  10967  ptcnp  21419  ptrescn  21436  itg2leub  23495  nmoubi  27611  nmopub  28751  nmfnleub  28768  nmcexi  28869  mblfinlem3  33428  ismblfin  33430  itg2addnc  33444  hbtlem2  37520
 Copyright terms: Public domain W3C validator