Theorem nfsbd 2111

Description: Deduction version of nfsb 2109. (Contributed by NM, 15-Feb-2013.)

Hypotheses

Ref	Expression
nfsbd.1	⊢ Ⅎxφ
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

Step	Hyp	Ref	Expression
1		nfsbd.1	. . . 4 ⊢ Ⅎxφ
2		nfsbd.2	. . . 4 ⊢ (φ → Ⅎzψ)
3	1, 2	alrimi 1765	. . 3 ⊢ (φ → ∀xℲzψ)
4		nfsb4t 2080	. . 3 ⊢ (∀xℲzψ → (¬ ∀z z = y → Ⅎz[y / x]ψ))
5	3, 4	syl 15	. 2 ⊢ (φ → (¬ ∀z z = y → Ⅎz[y / x]ψ))
6		a16nf 2051	. 2 ⊢ (∀z z = y → Ⅎz[y / x]ψ)
7	5, 6	pm2.61d2 152	1 ⊢ (φ → Ⅎz[y / x]ψ)

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  2508