Metamath Proof Explorer


Theorem sb4bOLD

Description: Obsolete version of sb4b as of 21-Feb-2024. (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 25-Jul-2023) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion sb4bOLD ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 nfeqf2 ( ¬ ∀ 𝑥 𝑥 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
2 nfnf1 𝑥𝑥 𝑦 = 𝑡
3 id ( Ⅎ 𝑥 𝑦 = 𝑡 → Ⅎ 𝑥 𝑦 = 𝑡 )
4 2 3 nfan1 𝑥 ( Ⅎ 𝑥 𝑦 = 𝑡𝑦 = 𝑡 )
5 equequ2 ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦𝑥 = 𝑡 ) )
6 5 imbi1d ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦𝜑 ) ↔ ( 𝑥 = 𝑡𝜑 ) ) )
7 6 adantl ( ( Ⅎ 𝑥 𝑦 = 𝑡𝑦 = 𝑡 ) → ( ( 𝑥 = 𝑦𝜑 ) ↔ ( 𝑥 = 𝑡𝜑 ) ) )
8 4 7 albid ( ( Ⅎ 𝑥 𝑦 = 𝑡𝑦 = 𝑡 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )
9 8 pm5.74da ( Ⅎ 𝑥 𝑦 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) ) )
10 1 9 syl ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ↔ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) ) )
11 10 albidv ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) ) )
12 df-sb ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )
13 ax6ev 𝑦 𝑦 = 𝑡
14 13 a1bi ( ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )
15 19.23v ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) ↔ ( ∃ 𝑦 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )
16 14 15 bitr4i ( ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )
17 11 12 16 3bitr4g ( ¬ ∀ 𝑥 𝑥 = 𝑡 → ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡𝜑 ) ) )