Theorem hbsb 2110

Description: If z is not free in φ, it is not free in [y / x]φ when y and z are distinct. (Contributed by NM, 12-Aug-1993.)

Hypothesis

Ref	Expression
hbsb.1	⊢ (φ → ∀zφ)

Assertion

Ref	Expression
hbsb	⊢ ([y / x]φ → ∀z[y / x]φ)

Distinct variable group:   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		hbsb.1	. . . 4 ⊢ (φ → ∀zφ)
2	1	nfi 1551	. . 3 ⊢ Ⅎzφ
3	2	nfsb 2109	. 2 ⊢ Ⅎz[y / x]φ
4	3	nfri 1762	1 ⊢ ([y / x]φ → ∀z[y / x]φ)

Syntax hints:   → wi 4  ∀wal 1540  [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:  hbab  2344  hblem  2458