MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfriotadw Structured version   Visualization version   GIF version

Theorem nfriotadw 7396
Description: Deduction version of nfriota 7400 with a disjoint variable condition, which contrary to nfriotad 7399 does not require ax-13 2375. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2375. (Revised by GG, 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 7388 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadw.1 . . . . . 6 𝑦𝜑
3 nfnaew 2147 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1897 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvd 2904 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 481 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotadw.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2915 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotadw.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 480 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1895 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotadw 6519 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 412 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 6518 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 biidd 262 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1716drnf1v 2373 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
1817albidv 1918 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓))))
19 df-nfc 2890 . . . . 5 (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑥 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
20 df-nfc 2890 . . . . 5 (𝑦(℩𝑦(𝑦𝐴𝜓)) ↔ ∀𝑤𝑦 𝑤 ∈ (℩𝑦(𝑦𝐴𝜓)))
2118, 19, 203bitr4g 314 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
2215, 21mpbiri 258 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
2314, 22pm2.61d2 181 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
241, 23nfcxfrd 2902 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wnf 1780  wcel 2106  wnfc 2888  cio 6514  crio 7387
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-sn 4632  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  nfriota  7400
  Copyright terms: Public domain W3C validator