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Theorem nfrald 3360
Description: Deduction version of nfral 3362. Usage of this theorem is discouraged because it depends on ax-13 2363. Use the weaker nfraldw 3298 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 3054 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfrald.1 . . 3 𝑦𝜑
3 nfcvf 2924 . . . . . 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 2906 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 480 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfimd 1889 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfald2 2436 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1848 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1531  wnf 1777  wcel 2098  wnfc 2875  wral 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054
This theorem is referenced by:  nfrexd  3361  nfral  3362
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