This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb6f

Description: Equivalence for substitution when y is not free in ph . The implication "to the left" is sb2 and does not require the nonfreeness hypothesis. Theorem sb6 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis sb6f.1 ⊢ Ⅎ 𝑦 𝜑
Assertion sb6f ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 sb6f.1 ⊢ Ⅎ 𝑦 𝜑
2 1 nf5ri ⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3 2 sbimi ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 )
4 sb4a ⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
5 3 4 syl ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
6 sb2 ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
7 5 6 impbii ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )