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Mirrors > Home > MPE Home > Th. List > nfiotadw | Structured version Visualization version GIF version |
Description: Deduction version of nfiotaw 6311. Version of nfiotad 6312 with a disjoint variable condition, which does not require ax-13 2389. (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 6308 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotadw.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotadw.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfvd 1915 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 4, 5 | nfbid 1902 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
7 | 3, 6 | nfald 2346 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
8 | 2, 7 | nfabdw 2999 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
9 | 8 | nfunid 4837 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
10 | 1, 9 | nfcxfrd 2975 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 Ⅎwnf 1783 {cab 2798 Ⅎwnfc 2960 ∪ cuni 4831 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-v 3493 df-in 3936 df-ss 3945 df-sn 4561 df-uni 4832 df-iota 6307 |
This theorem is referenced by: nfiotaw 6311 nfriotadw 7115 |
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