Theorem sbnv 1765

Description: Version of sbn 1823 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)

Assertion

Ref	Expression
sbnv	⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem sbnv

Step	Hyp	Ref	Expression
1		sb6 1763	. . 3 ⊢ ([y / x] ¬ φ ↔ ∀x(x = y → ¬ φ))
2		alinexa 1491	. . 3 ⊢ (∀x(x = y → ¬ φ) ↔ ¬ ∃x(x = y ∧ φ))
3	1, 2	bitri 173	. 2 ⊢ ([y / x] ¬ φ ↔ ¬ ∃x(x = y ∧ φ))
4		sb5 1764	. 2 ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ))
5	3, 4	xchbinxr 607	1 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-sb 1643

This theorem is referenced by:  sbn  1823