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Theorem nfsbd 2566
Description: Deduction version of nfsb 2567. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 15-Feb-2013.) (New usage is discouraged.)
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 2215 . . 3 (𝜑 → ∀𝑥𝑧𝜓)
4 nfsb4t 2541 . . 3 (∀𝑥𝑧𝜓 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
53, 4syl 17 . 2 (𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
6 axc16nf 2266 . 2 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
75, 6pm2.61d2 184 1 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  wnf 1785  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071
This theorem is referenced by:  nfsb  2567  nfabd  3004  wl-sb8eut  34890
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