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Metamath Proof Explorer

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