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Theorem bj-reabeq 37028
Description: Relative form of eqabb 2881. (Contributed by BJ, 27-Dec-2023.)
Assertion
Ref Expression
bj-reabeq ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-reabeq
StepHypRef Expression
1 dfrab3 4319 . . 3 {𝑥𝑉𝜑} = (𝑉 ∩ {𝑥𝜑})
21eqeq2i 2750 . 2 ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ (𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}))
3 nfcv 2905 . . 3 𝑥𝐴
4 nfab1 2907 . . 3 𝑥{𝑥𝜑}
5 nfcv 2905 . . 3 𝑥𝑉
63, 4, 5bj-rcleqf 37026 . 2 ((𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥 ∈ {𝑥𝜑}))
7 abid 2718 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
87bibi2i 337 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
98ralbii 3093 . 2 (∀𝑥𝑉 (𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
102, 6, 93bitri 297 1 ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  cin 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-in 3958
This theorem is referenced by: (None)
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