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Theorem ssrabeq 3911
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 3908 . . 3 {𝑥𝑉𝜑} ⊆ 𝑉
21biantru 525 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
3 eqss 3836 . 2 (𝑉 = {𝑥𝑉𝜑} ↔ (𝑉 ⊆ {𝑥𝑉𝜑} ∧ {𝑥𝑉𝜑} ⊆ 𝑉))
42, 3bitr4i 270 1 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  {crab 3094  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-in 3799  df-ss 3806
This theorem is referenced by:  difrab0eq  4262
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