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Theorem nfriotad 7119
 Description: Deduction version of nfriota 7120. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker nfriotadw 7116 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
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 7108 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotad.1 . . . . . 6 𝑦𝜑
3 nfnae 2452 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1896 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvf 3007 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 484 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotad.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 483 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2989 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 483 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1894 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotad 6313 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 415 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 6310 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 eqidd 2822 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦𝐴𝜓)) = (℩𝑦(𝑦𝐴𝜓)))
1716drnfc1 2997 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
1815, 17mpbiri 260 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
1914, 18pm2.61d2 183 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
201, 19nfcxfrd 2976 1 (𝜑𝑥(𝑦𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1531  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961  ℩cio 6306  ℩crio 7107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-sn 4561  df-uni 4832  df-iota 6308  df-riota 7108 This theorem is referenced by: (None)
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