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Metamath Proof Explorer
Description: One direction of eqabb is provable from fewer axioms. (Contributed by Wolf Lammen, 13-Feb-2025)
|
|
Ref |
Expression |
|
Assertion |
eqab |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → 𝐴 = { 𝑥 ∣ 𝜑 } ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
abid1 |
⊢ 𝐴 = { 𝑥 ∣ 𝑥 ∈ 𝐴 } |
| 2 |
|
abbi |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑥 ∣ 𝜑 } ) |
| 3 |
1 2
|
eqtrid |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → 𝐴 = { 𝑥 ∣ 𝜑 } ) |