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Theorem nfsbd 2557
Description: Deduction version of nfsb 2558. (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 2203 . . 3 (𝜑 → ∀𝑥𝑧𝜓)
4 nfsb4t 2532 . . 3 (∀𝑥𝑧𝜓 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
53, 4syl 17 . 2 (𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
6 axc16nf 2254 . 2 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
75, 6pm2.61d2 182 1 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526  wnf 1775  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  nfsb  2558  nfabd  2998  nfabd2OLD  3000  wl-sb8eut  34694
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