Metamath Proof Explorer
Description: Elimination of an existential quantifier, using implicit substitution.
(Contributed by NM, 2-Mar-1995)
|
|
Ref |
Expression |
|
Hypotheses |
ceqsexv.1 |
⊢ 𝐴 ∈ V |
|
|
ceqsexv.2 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ceqsexv.1 |
⊢ 𝐴 ∈ V |
2 |
|
ceqsexv.2 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
3 |
|
nfv |
⊢ Ⅎ 𝑥 𝜓 |
4 |
3 1 2
|
ceqsex |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 ) |