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Theorem nfrald 3365
Description: Deduction version of nfral 3367. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker nfraldw 3303 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrald.1 𝑦𝜑
nfrald.2 (𝜑𝑥𝐴)
nfrald.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrald (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Proof of Theorem nfrald
StepHypRef Expression
1 df-ral 3059 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfrald.1 . . 3 𝑦𝜑
3 nfcvf 2929 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 481 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfrald.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 480 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2911 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 480 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfimd 1890 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfald2 2440 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1849 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1532  wnf 1778  wcel 2099  wnfc 2879  wral 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059
This theorem is referenced by:  nfrexd  3366  nfral  3367
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