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Theorem nfiotad 6015
Description: Deduction version of nfiota 6016. (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 6013 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1992 . . . 4 𝑧𝜑
3 nfiotad.1 . . . . 5 𝑦𝜑
4 nfiotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
54adantr 472 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6 nfeqf1 2444 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
76adantl 473 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
85, 7nfbid 1981 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
93, 8nfald2 2471 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
102, 9nfabd 2923 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1110nfunid 4595 . 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 196  wa 383  wal 1630  wnf 1857  {cab 2746  wnfc 2889   cuni 4588  cio 6010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-sn 4322  df-uni 4589  df-iota 6012
This theorem is referenced by:  nfiota  6016  nfriotad  6782
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