Theorem nfsb4 2081

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
nfsb4.1	⊢ Ⅎzφ

Assertion

Ref	Expression
nfsb4	⊢ (¬ ∀z z = y → Ⅎz[y / x]φ)

Proof of Theorem nfsb4

Step	Hyp	Ref	Expression
1		nfsb4t 2080	. 2 ⊢ (∀xℲzφ → (¬ ∀z z = y → Ⅎz[y / x]φ))
2		nfsb4.1	. 2 ⊢ Ⅎzφ
3	1, 2	mpg 1548	1 ⊢ (¬ ∀z z = y → Ⅎz[y / x]φ)

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:  sbco2  2086  nfsb  2109  sbal1  2126