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Mirrors > Home > MPE Home > Th. List > nfiotadw | Structured version Visualization version GIF version |
Description: Deduction version of nfiotaw 6318. Version of nfiotad 6319 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 18-Feb-2013.) (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 6315 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotadw.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotadw.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfvd 1916 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 4, 5 | nfbid 1903 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
7 | 3, 6 | nfald 2347 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
8 | 2, 7 | nfabdw 3000 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
9 | 8 | nfunid 4844 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
10 | 1, 9 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 Ⅎwnf 1784 {cab 2799 Ⅎwnfc 2961 ∪ cuni 4838 ℩cio 6312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-in 3943 df-ss 3952 df-sn 4568 df-uni 4839 df-iota 6314 |
This theorem is referenced by: nfiotaw 6318 nfriotadw 7122 |
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