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Metamath Proof Explorer

Theorem dfsb2

Description: An alternate definition of proper substitution that, like dfsb1 , mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 17-Feb-2005) (New usage is discouraged.)

Ref Expression
Assertion dfsb2 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 sp ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 )
2 sbequ2 ⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 ) )
3 2 sps ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 ) )
4 orc ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
5 1 3 4 syl6an ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
6 sb4b ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
7 olc ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
8 6 7 biimtrdi ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
9 5 8 pm2.61i ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
10 sbequ1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )
11 10 imp ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
12 sb2 ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
13 11 12 jaoi ⊢ ( ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → [ 𝑦 / 𝑥 ] 𝜑 )
14 9 13 impbii ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ∨ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )