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Theorem sb9 2538
 Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2539. (Revised by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
Assertion
Ref Expression
sb9 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb9
StepHypRef Expression
1 sbequ12a 2253 . . . . 5 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
21equcoms 2027 . . . 4 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
32sps 2182 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43dral1 2450 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
5 nfnae 2445 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
6 nfnae 2445 . . 3 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
7 nfsb2 2501 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
87naecoms 2440 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
9 nfsb2 2501 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
102a1i 11 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
115, 6, 8, 9, 10cbv2 2412 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
124, 11pm2.61i 185 1 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  sb9i  2539
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