Theorem nfsb4 2390

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 14-May-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 2389	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2		nfsb4.1	. 2 ⊢ Ⅎ𝑧𝜑
3	1, 2	mpg 1724	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco2  2415  nfsb  2440