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Theorem axrep6g 5289
Description: axrep6 5287 in class notation. It is equivalent to both ax-rep 5278 and abrexexg 7986, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
Assertion
Ref Expression
axrep6g ((𝐴𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem axrep6g
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 3321 . . . . . 6 (𝑧 = 𝐴 → (∃𝑥𝑧 𝜓 ↔ ∃𝑥𝐴 𝜓))
21abbidv 2807 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥𝑧 𝜓} = {𝑦 ∣ ∃𝑥𝐴 𝜓})
32eleq1d 2825 . . . 4 (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V))
43imbi2d 340 . . 3 (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)))
5 axrep6 5287 . . . 4 (∀𝑥∃*𝑦𝜓 → ∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓))
6 abbi 2806 . . . . . 6 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦𝑦𝑤} = {𝑦 ∣ ∃𝑥𝑧 𝜓})
7 abid2 2878 . . . . . . 7 {𝑦𝑦𝑤} = 𝑤
8 vex 3483 . . . . . . 7 𝑤 ∈ V
97, 8eqeltri 2836 . . . . . 6 {𝑦𝑦𝑤} ∈ V
106, 9eqeltrrdi 2849 . . . . 5 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
1110exlimiv 1929 . . . 4 (∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
125, 11syl 17 . . 3 (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
134, 12vtoclg 3553 . 2 (𝐴𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V))
1413imp 406 1 ((𝐴𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107  ∃*wmo 2537  {cab 2713  wrex 3069  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481
This theorem is referenced by:  funimaexg  6652  abrexexg  7986
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