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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfintd | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| nfintd.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| Ref | Expression | 
|---|---|
| nfintd | ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-int 4946 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)} | |
| 2 | nfv 1913 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
| 4 | nfintd.1 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | 4 | nfcrd 2898 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴) | 
| 6 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | 
| 8 | 5, 7 | nfimd 1893 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) | 
| 9 | 3, 8 | nfald 2327 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) | 
| 10 | 2, 9 | nfabdw 2926 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}) | 
| 11 | 1, 10 | nfcxfrd 2903 | 1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 ∩ cint 4945 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-int 4946 | 
| This theorem is referenced by: (None) | 
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