Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
||
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 4411 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)} | |
2 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝑧) |
6 | nfintd.1 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
7 | 5, 6 | nfeld 2759 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴) |
8 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
10 | 7, 9 | nfimd 1812 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
11 | 3, 10 | nfald 2151 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)) |
12 | 2, 11 | nfabd 2771 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}) |
13 | 1, 12 | nfcxfrd 2750 | 1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-int 4411 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |