Theorem sbn 1945

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 1881	. . . 4 ⊢ ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
2	1	sbbii 1758	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑)
3		sbnv 1881	. . 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 1519	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
6	5	hbn 1647	. . 3 ⊢ (¬ 𝜑 → ∀𝑧 ¬ 𝜑)
7	6	sbco2vh 1938	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑)
8	5	sbco2vh 1938	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9	8	notbii 663	. 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 1755

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbcng  2995  difab  3396  rabeq0  3444  abeq0  3445  ssfirab  6911