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Theorem nfsbd 2111
 Description: Deduction version of nfsb 2109. (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 1765 . . 3 (φxzψ)
4 nfsb4t 2080 . . 3 (xzψ → (¬ z z = y → Ⅎz[y / x]ψ))
53, 4syl 15 . 2 (φ → (¬ z z = y → Ⅎz[y / x]ψ))
6 a16nf 2051 . 2 (z z = y → Ⅎz[y / x]ψ)
75, 6pm2.61d2 152 1 (φ → Ⅎz[y / x]ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  nfabd2  2507
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