Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | reubidva.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| Assertion | reubidva | ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reubidva.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | 1 | pm5.32da | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) ) |
| 3 | 2 | eubidv | ⊢ ( 𝜑 → ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) ) |
| 4 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) | |
| 5 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜒 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) | |
| 6 | 3 4 5 | 3bitr4g | ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) ) |