Theorem sb9 2527

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2528. (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

Step	Hyp	Ref	Expression
1		sbequ12a 2255	. . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	equcoms 2019	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	2	sps 2186	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	dral1 2447	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
5		nfnae 2442	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
6		nfnae 2442	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
7		nfsb2 2491	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
8	7	naecoms 2437	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
9		nfsb2 2491	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
10	2	a1i 11	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
11	5, 6, 8, 9, 10	cbv2 2411	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
12	4, 11	pm2.61i 182	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  sb9i  2528