![]() |
Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfiund | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) |
Ref | Expression |
---|---|
nfiund.1 | ⊢ Ⅎ𝑥𝜑 |
nfiund.2 | ⊢ (𝜑 → Ⅎ𝑦𝐴) |
nfiund.3 | ⊢ (𝜑 → Ⅎ𝑦𝐵) |
Ref | Expression |
---|---|
nfiund | ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4757 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfv 1957 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiund.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfiund.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
5 | nfiund.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵) | |
6 | 5 | nfcrd 2927 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfrexd 3187 | . . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2954 | . 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2933 | 1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1827 ∈ wcel 2107 {cab 2763 Ⅎwnfc 2919 ∃wrex 3091 ∪ ciun 4755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-iun 4757 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |