Theorem sb9 2552

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2405. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2553. (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 2291	. . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	equcoms 2042	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	2	sps 2222	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	dral1 2472	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
5		nfnae 2467	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
6		nfnae 2467	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
7		nfsb2 2516	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
8	7	naecoms 2462	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦[𝑥 / 𝑦]𝜑)
9		nfsb2 2516	. . 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 2436	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
12	4, 11	pm2.61i 183	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1560  Ⅎwnf 1805  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093

This theorem is referenced by:  sb9i  2553