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Metamath Proof Explorer

Theorem nfsb4t

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 ). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 7-Apr-2004) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

Ref Expression
Assertion nfsb4t ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step Hyp Ref Expression
1 sbequ12 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
2 1 sps ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
3 2 drnf2 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 ↔ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
4 3 biimpd ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
5 4 spsd ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
6 5 impcom ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
7 6 a1d ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
8 nfnf1 ⊢ Ⅎ 𝑧 Ⅎ 𝑧 𝜑
9 8 nfal ⊢ Ⅎ 𝑧 ∀ 𝑥 Ⅎ 𝑧 𝜑
10 nfnae ⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
11 9 10 nfan ⊢ Ⅎ 𝑧 ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
12 nfa1 ⊢ Ⅎ 𝑥 ∀ 𝑥 Ⅎ 𝑧 𝜑
13 nfnae ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
14 12 13 nfan ⊢ Ⅎ 𝑥 ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
15 sp ⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 𝜑 )
16 15 adantr ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜑 )
17 nfsb2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
18 17 adantl ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
19 1 a1i ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) ) )
20 11 14 16 18 19 dvelimdf ⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
21 7 20 pm2.61dan ⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )