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Theorem elintab 3626
 Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1 𝐴 ∈ V
Assertion
Ref Expression
elintab (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3 𝐴 ∈ V
21elint 3621 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦))
3 nfsab1 2030 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1421 . . . 4 𝑥 𝐴𝑦
53, 4nfim 1464 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦)
6 nfv 1421 . . 3 𝑦(𝜑𝐴𝑥)
7 eleq1 2100 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2028 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 185 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
10 eleq2 2101 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
119, 10imbi12d 223 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ (𝜑𝐴𝑥)))
125, 6, 11cbval 1637 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ ∀𝑥(𝜑𝐴𝑥))
132, 12bitri 173 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  {cab 2026  Vcvv 2557  ∩ cint 3615 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-v 2559  df-int 3616 This theorem is referenced by:  elintrab  3627  intmin4  3643  intab  3644  intid  3960
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