Theorem sbnv 1937

Description: Version of sbn 2005 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 1935	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2		alinexa 1651	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3	1, 2	bitri 184	. 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
4		sb5 1936	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
5	3, 4	xchbinxr 689	1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395  ∃wex 1540  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-sb 1811

This theorem is referenced by:  sbn  2005