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Theorem nfrald 3083
 Description: Deduction version of nfral 3084. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 3056 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfrald.1 . . 3 𝑦𝜑
3 nfcvf 2927 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 473 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfrald.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 472 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2912 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 472 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfimd 1973 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfald2 2472 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1929 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1630  Ⅎwnf 1857   ∈ wcel 2140  Ⅎwnfc 2890  ∀wral 3051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056 This theorem is referenced by:  nfral  3084  nfrexd  3145
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