Theorem sbnv 1816

Description: Version of sbn 1874 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 1814	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2		alinexa 1539	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3	1, 2	bitri 182	. 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
4		sb5 1815	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
5	3, 4	xchbinxr 643	1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1287  ∃wex 1426  [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-sb 1693

This theorem is referenced by:  sbn  1874