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Theorem nfriotadw 7321
Description: Deduction version of nfriota 7326 with a disjoint variable condition, which contrary to nfriotad 7325 does not require ax-13 2370. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
nfriotadw.1 𝑦𝜑
nfriotadw.2 (𝜑 → Ⅎ𝑥𝜓)
nfriotadw.3 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfriotadw (𝜑𝑥(𝑦𝐴 𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfriotadw
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-riota 7313 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadw.1 . . . . . 6 𝑦𝜑
3 nfnaew 2145 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1902 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvd 2908 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 482 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotadw.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2918 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotadw.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 481 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1900 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotadw 6451 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 413 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 6450 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 biidd 261 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1716drnf1v 2368 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1817albidv 1923 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
19 df-nfc 2889 . . . . 5 (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
20 df-nfc 2889 . . . . 5 (𝑦(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
2118, 19, 203bitr4g 313 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
2215, 21mpbiri 257 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
2314, 22pm2.61d2 181 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
241, 23nfcxfrd 2906 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1539  wnf 1785  wcel 2106  wnfc 2887  cio 6446  crio 7312
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-v 3447  df-in 3917  df-ss 3927  df-sn 4587  df-uni 4866  df-iota 6448  df-riota 7313
This theorem is referenced by:  nfriota  7326
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