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Theorem axrep6g 5251
Description: axrep6 5250 in class notation. It is equivalent to both ax-rep 5243 and abrexexg 7894, 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 3309 . . . . . 6 (𝑧 = 𝐴 → (∃𝑥𝑧 𝜓 ↔ ∃𝑥𝐴 𝜓))
21abbidv 2802 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥𝑧 𝜓} = {𝑦 ∣ ∃𝑥𝐴 𝜓})
32eleq1d 2819 . . . 4 (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V))
43imbi2d 341 . . 3 (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)))
5 axrep6 5250 . . . 4 (∀𝑥∃*𝑦𝜓 → ∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓))
6 abbi 2801 . . . . . 6 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦𝑦𝑤} = {𝑦 ∣ ∃𝑥𝑧 𝜓})
7 abid2 2872 . . . . . . 7 {𝑦𝑦𝑤} = 𝑤
8 vex 3448 . . . . . . 7 𝑤 ∈ V
97, 8eqeltri 2830 . . . . . 6 {𝑦𝑦𝑤} ∈ V
106, 9eqeltrrdi 2843 . . . . 5 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
1110exlimiv 1934 . . . 4 (∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
125, 11syl 17 . . 3 (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
134, 12vtoclg 3524 . 2 (𝐴𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V))
1413imp 408 1 ((𝐴𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃*wmo 2533  {cab 2710  wrex 3070  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-rep 5243
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-v 3446
This theorem is referenced by:  funimaexg  6588  abrexexg  7894
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