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Theorem ralab 2763
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 2358 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒))
2 vex 2615 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 2748 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54imbi1i 236 . . 3 ((𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ (𝜓𝜒))
65albii 1400 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ ∀𝑥(𝜓𝜒))
71, 6bitri 182 1 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wcel 1434  {cab 2069  wral 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614
This theorem is referenced by:  funcnvuni  5036  ralrnmpt2  5694  pitonn  7288
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