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Theorem ssrabeq 4078
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 4073 . . 3 {𝑥𝑉𝜑} ⊆ 𝑉
21biantru 530 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
3 eqss 3993 . 2 (𝑉 = {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
42, 3bitr4i 277 1 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  {crab 3431  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3951  df-ss 3961
This theorem is referenced by:  difrab0eq  4465
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