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Theorem nfriotadw 7178
Description: Deduction version of nfriota 7183 with a disjoint variable condition, which contrary to nfriotad 7182 does not require ax-13 2371. (Contributed by NM, 18-Feb-2013.) (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 7170 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadw.1 . . . . . 6 𝑦𝜑
3 nfnaew 2149 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1907 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvd 2905 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 485 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotadw.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2915 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotadw.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 484 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1905 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotadw 6341 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 416 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 6340 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 biidd 265 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1716drnf1v 2370 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1817albidv 1928 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
19 df-nfc 2886 . . . . 5 (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
20 df-nfc 2886 . . . . 5 (𝑦(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
2118, 19, 203bitr4g 317 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
2215, 21mpbiri 261 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
2314, 22pm2.61d2 184 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
241, 23nfcxfrd 2903 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1541  wnf 1791  wcel 2110  wnfc 2884  cio 6336  crio 7169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883  df-sn 4542  df-uni 4820  df-iota 6338  df-riota 7170
This theorem is referenced by:  nfriota  7183
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