Theorem nfsb2 2482

Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
nfsb2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb2

Step	Hyp	Ref	Expression
1		nfna1 2149	. 2 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		hbsb2 2481	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
3	1, 2	nf5d 2280	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  Ⅎwnf 1785  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by:  nfsb4t  2498  sbco3  2512  sb9  2518  wl-nfs1t  36394