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Theorem nfiotad 6499
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.)
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 6495 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1915 . . . 4 𝑧𝜑
3 nfiotad.1 . . . . 5 𝑦𝜑
4 nfiotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
54adantr 479 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6 nfeqf1 2376 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
76adantl 480 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
85, 7nfbid 1903 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
93, 8nfald2 2442 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
102, 9nfabd 2926 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1110nfunid 4913 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
121, 11nfcxfrd 2900 1 (𝜑𝑥(℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1537  wnf 1783  {cab 2707  wnfc 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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