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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.
StepHypRef Expression
1 sbnv 1817 . . . 4 ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
21sbbii 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)
42, 3bitri 183 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
5 ax-17 1465 . . . 4 (𝜑 → ∀𝑧𝜑)
65hbn 1590 . . 3 𝜑 → ∀𝑧 ¬ 𝜑)
76sbco2v 1870 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑)
85sbco2v 1870 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
98notbii 630 . 2 (¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
104, 7, 93bitr3i 209 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
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
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