Theorem nfsb2 2515

Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2404. Use nfs1v 2191 when sufficient. (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 2187	. 2 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		hbsb2 2514	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
3	1, 2	nf5d 2319	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by:  nfsb4t  2531  sbco3  2545  sb9  2551  wl-nfs1t  38212