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Mirrors > Home > MPE Home > Th. List > nfiotad | Structured version Visualization version GIF version |
Description: Deduction version of nfiota 5855. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfiotad.1 | ⊢ Ⅎ𝑦𝜑 |
nfiotad.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfiotad | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiota2 5852 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotad.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotad.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
6 | nfeqf1 2299 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | |
7 | 6 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧) |
8 | 5, 7 | nfbid 1832 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
9 | 3, 8 | nfald2 2331 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
10 | 2, 9 | nfabd 2785 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
11 | 10 | nfunid 4443 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
12 | 1, 11 | nfcxfrd 2763 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 Ⅎwnf 1708 {cab 2608 Ⅎwnfc 2751 ∪ cuni 4436 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-sn 4178 df-uni 4437 df-iota 5851 |
This theorem is referenced by: nfiota 5855 nfriotad 6619 |
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