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Theorem ssrabeq 3026
Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
ssrabeq (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
Distinct variable group:   𝑥,𝑉
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrabeq
StepHypRef Expression
1 ssrab2 3025 . . 3 {𝑥𝑉𝜑} ⊆ 𝑉
21biantru 286 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
3 eqss 2960 . 2 (𝑉 = {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
42, 3bitr4i 176 1 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98   = wceq 1243  {crab 2310  wss 2917
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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924  df-ss 2931
This theorem is referenced by:  difrab0eqim  3288
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