Theorem sbn 1875

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 1817	. . . 4 ⊢ ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
2	1	sbbii 1696	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑)
3		sbnv 1817	. . 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 1465	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
6	5	hbn 1590	. . 3 ⊢ (¬ 𝜑 → ∀𝑧 ¬ 𝜑)
7	6	sbco2v 1870	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑)
8	5	sbco2v 1870	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9	8	notbii 630	. 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 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694

This theorem is referenced by:  sbcng  2882  difab  3271  rabeq0  3318  abeq0  3319  ssfirab  6699