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