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 Description: Deduction version of nfiota 6309. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfiotadw 6306 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfiotad.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression

Proof of Theorem nfiotad
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6304 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1916 . . . 4 𝑧𝜑
3 nfiotad.1 . . . . 5 𝑦𝜑
4 nfiotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
54adantr 484 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6 nfeqf1 2399 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
76adantl 485 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
85, 7nfbid 1904 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
93, 8nfald2 2469 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
102, 9nfabd 3004 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1110nfunid 4831 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
121, 11nfcxfrd 2981 1 (𝜑𝑥(℩𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785  {cab 2802  Ⅎwnfc 2962  ∪ cuni 4825  ℩cio 6301 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3483  df-in 3927  df-ss 3937  df-sn 4552  df-uni 4826  df-iota 6303 This theorem is referenced by:  nfiota  6309  nfriotad  7119
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