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Theorem ralab 3561
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 3094 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒))
2 vex 3388 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 3542 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54imbi1i 341 . . 3 ((𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ (𝜓𝜒))
65albii 1915 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ ∀𝑥(𝜓𝜒))
71, 6bitri 267 1 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wcel 2157  {cab 2785  wral 3089
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-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387
This theorem is referenced by:  ralrnmpt2  7009  funcnvuni  7354  kardex  9007  karden  9008  fimaxre3  11262  ptcnp  21754  ptrescn  21771  itg2leub  23842  nmoubi  28152  nmopub  29292  nmfnleub  29309  nmcexi  29410  mblfinlem3  33937  ismblfin  33939  itg2addnc  33952  hbtlem2  38479
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