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Theorem nfrald 3126
Description: Deduction version of nfral 3127. (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 3095 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfrald.1 . . 3 𝑦𝜑
3 nfcvf 2960 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 475 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfrald.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 474 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2943 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 474 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfimd 1940 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfald2 2411 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1898 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wal 1599  wnf 1827  wcel 2107  wnfc 2919  wral 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095
This theorem is referenced by:  nfral  3127  nfrexd  3187
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