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Theorem sb9 1904
Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Assertion
Ref Expression
sb9 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb9
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sb9v 1903 . . 3 (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑)
2 sbcom 1898 . . . 4 ([𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑤 / 𝑦]𝜑)
32albii 1405 . . 3 (∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑)
4 sb9v 1903 . . 3 (∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑)
51, 3, 43bitri 205 . 2 (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑)
6 ax-17 1465 . . . 4 (𝜑 → ∀𝑤𝜑)
76sbco2h 1887 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
87albii 1405 . 2 (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
96sbco2h 1887 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑)
109albii 1405 . 2 (∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑦]𝜑)
115, 8, 103bitr3ri 210 1 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1288  [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-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-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694
This theorem is referenced by:  sb9i  1905
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