Description: Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Alexander van der Vekens, 17-Jun-2017) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | sb8eu.1 | ⊢ Ⅎ 𝑦 𝜑 | |
Assertion | sb8mo | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8eu.1 | ⊢ Ⅎ 𝑦 𝜑 | |
2 | 1 | sb8e | ⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) |
3 | 1 | sb8eu | ⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) |
4 | 2 3 | imbi12i | ⊢ ( ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) ) |
5 | moeu | ⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ) | |
6 | moeu | ⊢ ( ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) ) | |
7 | 4 5 6 | 3bitr4i | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) |