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Mirrors > Home > MPE Home > Th. List > nfiotadw | Structured version Visualization version GIF version |
Description: Deduction version of nfiotaw 6499. Version of nfiotad 6500 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
nfiotadw.1 | ⊢ Ⅎ𝑦𝜑 |
nfiotadw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfiotadw | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiota2 6496 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotadw.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotadw.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfvd 1918 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 4, 5 | nfbid 1905 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
7 | 3, 6 | nfald 2321 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
8 | 2, 7 | nfabdw 2926 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
9 | 8 | nfunid 4914 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
10 | 1, 9 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 Ⅎwnf 1785 {cab 2709 Ⅎwnfc 2883 ∪ cuni 4908 ℩cio 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-v 3476 df-in 3955 df-ss 3965 df-sn 4629 df-uni 4909 df-iota 6495 |
This theorem is referenced by: nfiotaw 6499 nfriotadw 7375 |
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