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Theorem nfriotad 6604
Description: Deduction version of nfriota 6605. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfriotad.1 𝑦𝜑
nfriotad.2 (𝜑 → Ⅎ𝑥𝜓)
nfriotad.3 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfriotad (𝜑𝑥(𝑦𝐴 𝜓))

Proof of Theorem nfriotad
StepHypRef Expression
1 df-riota 6596 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotad.1 . . . . . 6 𝑦𝜑
3 nfnae 2316 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1826 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvf 2785 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 482 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotad.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2770 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 481 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1824 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotad 5842 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 450 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 5841 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 eqidd 2621 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦𝐴𝜓)) = (℩𝑦(𝑦𝐴𝜓)))
1716drnfc1 2779 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
1815, 17mpbiri 248 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
1914, 18pm2.61d2 172 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
201, 19nfcxfrd 2761 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1479  wnf 1706  wcel 1988  wnfc 2749  cio 5837  crio 6595
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  df-riota 6596
This theorem is referenced by:  nfriota  6605
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