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Theorem axrep6g 5286
Description: axrep6 5285 in class notation. It is equivalent to both ax-rep 5278 and abrexexg 7943, 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 3315 . . . . . 6 (𝑧 = 𝐴 → (∃𝑥𝑧 𝜓 ↔ ∃𝑥𝐴 𝜓))
21abbidv 2795 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥𝑧 𝜓} = {𝑦 ∣ ∃𝑥𝐴 𝜓})
32eleq1d 2812 . . . 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 5285 . . . 4 (∀𝑥∃*𝑦𝜓 → ∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓))
6 abbi 2794 . . . . . 6 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦𝑦𝑤} = {𝑦 ∣ ∃𝑥𝑧 𝜓})
7 abid2 2865 . . . . . . 7 {𝑦𝑦𝑤} = 𝑤
8 vex 3472 . . . . . . 7 𝑤 ∈ V
97, 8eqeltri 2823 . . . . . 6 {𝑦𝑦𝑤} ∈ V
106, 9eqeltrrdi 2836 . . . . 5 (∀𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
1110exlimiv 1925 . . . 4 (∃𝑤𝑦(𝑦𝑤 ↔ ∃𝑥𝑧 𝜓) → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
125, 11syl 17 . . 3 (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥𝑧 𝜓} ∈ V)
134, 12vtoclg 3537 . 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 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  ∃*wmo 2526  {cab 2703  wrex 3064  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-rep 5278
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-v 3470
This theorem is referenced by:  funimaexg  6627  abrexexg  7943
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