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Mirrors > Home > MPE Home > Th. List > nfiotad | Structured version Visualization version GIF version |
Description: Deduction version of nfiota 6500. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfiotadw 6497 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfiotad.1 | ⊢ Ⅎ𝑦𝜑 |
nfiotad.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfiotad | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiota2 6495 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotad.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotad.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 4 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
6 | nfeqf1 2376 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | |
7 | 6 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧) |
8 | 5, 7 | nfbid 1903 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
9 | 3, 8 | nfald2 2442 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
10 | 2, 9 | nfabd 2926 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
11 | 10 | nfunid 4913 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
12 | 1, 11 | nfcxfrd 2900 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1537 Ⅎwnf 1783 {cab 2707 Ⅎwnfc 2881 ∪ cuni 4907 ℩cio 6492 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-13 2369 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-v 3474 df-in 3954 df-ss 3964 df-sn 4628 df-uni 4908 df-iota 6494 |
This theorem is referenced by: nfiota 6500 nfriotad 7379 |
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