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Theorem nfsbd 2430
 Description: Deduction version of nfsb 2428. (Contributed by NM, 15-Feb-2013.)
Hypotheses
Ref Expression
nfsbd.1 𝑥𝜑
nfsbd.2 (𝜑 → Ⅎ𝑧𝜓)
Assertion
Ref Expression
nfsbd (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem nfsbd
StepHypRef Expression
1 nfsbd.1 . . . 4 𝑥𝜑
2 nfsbd.2 . . . 4 (𝜑 → Ⅎ𝑧𝜓)
31, 2alrimi 2069 . . 3 (𝜑 → ∀𝑥𝑧𝜓)
4 nfsb4t 2377 . . 3 (∀𝑥𝑧𝜓 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
53, 4syl 17 . 2 (𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
6 axc16nf 2122 . 2 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
75, 6pm2.61d2 171 1 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  Ⅎwnf 1699  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  nfabd2  2770  wl-sb8eut  32538
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