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Theorem ssrabeq 3190
 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 3189 . . 3 {𝑥𝑉𝜑} ⊆ 𝑉
21biantru 300 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
3 eqss 3119 . 2 (𝑉 = {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
42, 3bitr4i 186 1 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332  {crab 2422   ⊆ wss 3078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-rab 2427  df-in 3084  df-ss 3091 This theorem is referenced by:  difrab0eqim  3436
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