Theorem nfsb2 2058

Description: Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)

Assertion

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

Proof of Theorem nfsb2

Step	Hyp	Ref	Expression
1		nfnae 1956	. 2 ⊢ Ⅎx ¬ ∀x x = y
2		hbsb2 2057	. 2 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))
3	1, 2	nfd 1766	1 ⊢ (¬ ∀x x = y → Ⅎx[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:  sbequi  2059  nfsb4t  2080  sbco3  2088