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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 )