Theorem sbn 1923

Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)

Assertion

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

Proof of Theorem sbn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbnv 1860	. . . 4 ⊢ ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
2	1	sbbii 1738	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑)
3		sbnv 1860	. . 3 ⊢ ([𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
4	2, 3	bitri 183	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
5		ax-17 1506	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
6	5	hbn 1632	. . 3 ⊢ (¬ 𝜑 → ∀𝑧 ¬ 𝜑)
7	6	sbco2vh 1916	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑)
8	5	sbco2vh 1916	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9	8	notbii 657	. 2 ⊢ (¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
10	4, 7, 9	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbcng  2944  difab  3340  rabeq0  3387  abeq0  3388  ssfirab  6815