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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 4870 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)} | |
2 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfintd.1 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 4 | nfcrd 2969 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴) |
6 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
8 | 5, 7 | nfimd 1891 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
9 | 3, 8 | nfald 2343 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
10 | 2, 9 | nfabdw 3000 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}) |
11 | 1, 10 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 Ⅎwnf 1780 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∩ cint 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-int 4870 |
This theorem is referenced by: (None) |
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