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| 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.) | 
| Ref | Expression | 
|---|---|
| axrep6g | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexeq 3321 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓)) | |
| 2 | 1 | abbidv 2807 | . . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓}) | 
| 3 | 2 | eleq1d 2825 | . . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) | 
| 4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))) | 
| 5 | axrep6 5287 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜓 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓)) | |
| 6 | abbi 2806 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ 𝑦 ∈ 𝑤} = {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓}) | |
| 7 | abid2 2878 | . . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} = 𝑤 | |
| 8 | vex 3483 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
| 9 | 7, 8 | eqeltri 2836 | . . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} ∈ V | 
| 10 | 6, 9 | eqeltrrdi 2849 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) | 
| 11 | 10 | exlimiv 1929 | . . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) | 
| 12 | 5, 11 | syl 17 | . . 3 ⊢ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) | 
| 13 | 4, 12 | vtoclg 3553 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) | 
| 14 | 13 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 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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