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Theorem nfriotad 7139
Description: Deduction version of nfriota 7140. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfriotadw 7135 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 7127 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotad.1 . . . . . 6 𝑦𝜑
3 nfnae 2434 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
42, 3nfan 1906 . . . . 5 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5 nfcvf 2928 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
65adantl 485 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
7 nfriotad.3 . . . . . . . 8 (𝜑𝑥𝐴)
87adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
96, 8nfeld 2910 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
10 nfriotad.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
1110adantr 484 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
129, 11nfand 1904 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
134, 12nfiotad 6302 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥(℩𝑦(𝑦𝐴𝜓)))
1413ex 416 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓))))
15 nfiota1 6299 . . . 4 𝑦(℩𝑦(𝑦𝐴𝜓))
16 eqidd 2739 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦𝐴𝜓)) = (℩𝑦(𝑦𝐴𝜓)))
1716drnfc1 2918 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥(℩𝑦(𝑦𝐴𝜓)) ↔ 𝑦(℩𝑦(𝑦𝐴𝜓))))
1815, 17mpbiri 261 . . 3 (∀𝑥 𝑥 = 𝑦𝑥(℩𝑦(𝑦𝐴𝜓)))
1914, 18pm2.61d2 184 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
201, 19nfcxfrd 2898 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1540  wnf 1790  wcel 2114  wnfc 2879  cio 6295  crio 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2372  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-in 3850  df-ss 3860  df-sn 4517  df-uni 4797  df-iota 6297  df-riota 7127
This theorem is referenced by: (None)
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