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Theorem ralab 2701
 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 2311 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒))
2 vex 2560 . . . . 5 𝑥 ∈ V
3 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3elab 2687 . . . 4 (𝑥 ∈ {𝑦𝜑} ↔ 𝜓)
54imbi1i 227 . . 3 ((𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ (𝜓𝜒))
65albii 1359 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜒) ↔ ∀𝑥(𝜓𝜒))
71, 6bitri 173 1 (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  {cab 2026  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559 This theorem is referenced by:  funcnvuni  4968  ralrnmpt2  5615  pitonn  6924
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