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Metamath Proof Explorer

Theorem sb4b

Description: Simplified definition of substitution when variables are distinct. Version of sb6 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

Ref Expression
Assertion sb4b ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 nfna1 ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑡
2 nfeqf2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
3 1 2 nfan1 ⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 )
4 equequ2 ⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) )
5 4 imbi1d ⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
6 5 adantl ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
7 3 6 albid ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
8 7 pm5.74da ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
9 8 albidv ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
10 dfsb ⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
11 ax6ev ⊢ ∃ 𝑦 𝑦 = 𝑡
12 11 a1bi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
13 19.23v ⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
14 12 13 bitr4i ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
15 9 10 14 3bitr4g ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )