This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Restricted existential specialization in an equality, using implicit
substitution. (Contributed by BJ, 2-Sep-2022)
|
|
Ref |
Expression |
|
Hypothesis |
rspceeqv.1 |
⊢ ( 𝑥 = 𝐴 → 𝐶 = 𝐷 ) |
|
Assertion |
rspceeqv |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝐸 = 𝐶 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rspceeqv.1 |
⊢ ( 𝑥 = 𝐴 → 𝐶 = 𝐷 ) |
| 2 |
1
|
eqeq2d |
⊢ ( 𝑥 = 𝐴 → ( 𝐸 = 𝐶 ↔ 𝐸 = 𝐷 ) ) |
| 3 |
2
|
rspcev |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝐸 = 𝐶 ) |