Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | hbsb2 | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4b | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) | |
2 | sb2 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) | |
3 | 2 | axc4i | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) |
4 | 1 3 | syl6bi | ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) ) |