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Theorem nfiotad 6489
Description: Deduction version of nfiota 6490. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotadw 6487 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfiotad.1 𝑦𝜑
nfiotad.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfiotad (𝜑𝑥(℩𝑦𝜓))

Proof of Theorem nfiotad
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6485 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1917 . . . 4 𝑧𝜑
3 nfiotad.1 . . . . 5 𝑦𝜑
4 nfiotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
54adantr 481 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6 nfeqf1 2377 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
76adantl 482 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
85, 7nfbid 1905 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
93, 8nfald2 2443 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
102, 9nfabd 2927 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1110nfunid 4907 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
121, 11nfcxfrd 2901 1 (𝜑𝑥(℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539  wnf 1785  {cab 2708  wnfc 2882   cuni 4901  cio 6482
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-13 2370  ax-ext 2702
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 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3951  df-ss 3961  df-sn 4623  df-uni 4902  df-iota 6484
This theorem is referenced by:  nfiota  6490  nfriotad  7361
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