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Theorem ssrabeq 3997
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 3993 . . 3 {𝑥𝑉𝜑} ⊆ 𝑉
21biantru 533 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
3 eqss 3916 . 2 (𝑉 = {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
42, 3bitr4i 281 1 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  {crab 3065  wss 3866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883
This theorem is referenced by:  difrab0eq  4384
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