Theorem sb9

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) Allow a shortening of sb9i . (Revised by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbequ12a	⊢ ( 𝑦 = 𝑥 → ( [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
2	1	equcoms	⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
3	2	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	3	dral1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
5		nfnae	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
6		nfnae	⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦
7		nfsb2	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → Ⅎ 𝑦 [ 𝑥 / 𝑦 ] 𝜑 )
8	7	naecoms	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑦 [ 𝑥 / 𝑦 ] 𝜑 )
9		nfsb2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
10	2	a1i	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) ) )
11	5 6 8 9 10	cbv2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
12	4 11	pm2.61i	⊢ ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )