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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfiundg | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2365, see nfiund 47974 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfiundg.1 | ⊢ Ⅎ𝑥𝜑 |
nfiundg.2 | ⊢ (𝜑 → Ⅎ𝑦𝐴) |
nfiundg.3 | ⊢ (𝜑 → Ⅎ𝑦𝐵) |
Ref | Expression |
---|---|
nfiundg | ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4992 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiundg.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfiundg.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
5 | nfiundg.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵) | |
6 | 5 | nfcrd 2886 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfrexd 3363 | . . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2922 | . 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2896 | 1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1777 ∈ wcel 2098 {cab 2703 Ⅎwnfc 2877 ∃wrex 3064 ∪ ciun 4990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2365 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-iun 4992 |
This theorem is referenced by: (None) |
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