Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | hbsb2e | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4e | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) ) | |
2 | sb2 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) → [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 ) | |
3 | 2 | axc4i | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 ) |
4 | 1 3 | syl | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 ) |