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Theorem bj-reabeq 37009
Description: Relative form of eqabb 2878. (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 4324 . . 3 {𝑥𝑉𝜑} = (𝑉 ∩ {𝑥𝜑})
21eqeq2i 2747 . 2 ((𝑉𝐴) = {𝑥𝑉𝜑} ↔ (𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}))
3 nfcv 2902 . . 3 𝑥𝐴
4 nfab1 2904 . . 3 𝑥{𝑥𝜑}
5 nfcv 2902 . . 3 𝑥𝑉
63, 4, 5bj-rcleqf 37007 . 2 ((𝑉𝐴) = (𝑉 ∩ {𝑥𝜑}) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥 ∈ {𝑥𝜑}))
7 abid 2715 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
87bibi2i 337 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
98ralbii 3090 . 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 1536  wcel 2105  {cab 2711  wral 3058  {crab 3432  cin 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rab 3433  df-v 3479  df-in 3969
This theorem is referenced by: (None)
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