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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 4882 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)} | |
2 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfintd.1 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 4 | nfcrd 2896 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴) |
6 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
8 | 5, 7 | nfimd 1897 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
9 | 3, 8 | nfald 2322 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
10 | 2, 9 | nfabdw 2930 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}) |
11 | 1, 10 | nfcxfrd 2906 | 1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1786 ∈ wcel 2106 {cab 2715 Ⅎwnfc 2887 ∩ cint 4881 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-int 4882 |
This theorem is referenced by: (None) |
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