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Theorem nfiotad 5842
Description: Deduction version of nfiota 5843. (Contributed by NM, 18-Feb-2013.)
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 5840 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1841 . . . 4 𝑧𝜑
3 nfiotad.1 . . . . 5 𝑦𝜑
4 nfiotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
54adantr 481 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6 nfeqf1 2297 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
76adantl 482 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
85, 7nfbid 1830 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
93, 8nfald2 2329 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
102, 9nfabd 2782 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1110nfunid 4434 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
121, 11nfcxfrd 2761 1 (𝜑𝑥(℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wnf 1706  {cab 2606  wnfc 2749   cuni 4427  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-sn 4169  df-uni 4428  df-iota 5839
This theorem is referenced by:  nfiota  5843  nfriotad  6604
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