Theorem sb9 2539

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2540. (Revised by Wolf Lammen, 15-Jun-2019.)

Assertion

Ref	Expression
sb9	⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb9

Step	Hyp	Ref	Expression
1		sbequ12a 2280	. . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	equcoms 2119	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	2	sps 2219	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	dral1 2423	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
5		nfnae 2418	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
6		nfnae 2418	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
7		nfsb2 2455	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
8	7	naecoms 2413	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
9		nfsb2 2455	. . 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 2386	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
12	4, 11	pm2.61i 177	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1651  Ⅎwnf 1879  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065

This theorem is referenced by:  sb9i  2540