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Theorem bj-reabeq 34744
 Description: Relative form of abeq2 2884. (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 4212 . . 3 {𝑥𝑉𝜑} = (𝑉 ∩ {𝑥𝜑})
21eqeq2i 2771 . 2 ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ (𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}))
3 nfcv 2919 . . 3 𝑥𝐴
4 nfab1 2921 . . 3 𝑥{𝑥𝜑}
5 nfcv 2919 . . 3 𝑥𝑉
63, 4, 5bj-rcleqf 34742 . 2 ((𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥 ∈ {𝑥𝜑}))
7 abid 2739 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
87bibi2i 341 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
98ralbii 3097 . 2 (∀𝑥𝑉 (𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
102, 6, 93bitri 300 1 ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  {crab 3074   ∩ cin 3857 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-in 3865 This theorem is referenced by: (None)
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