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Metamath Proof Explorer
Description: Equality theorem for class substitution. Class version of sbequ12 .
(Contributed by NM, 26-Sep-2003)
|
|
Ref |
Expression |
|
Assertion |
sbceq1a |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sbid |
⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 ) |
| 2 |
|
dfsbcq2 |
⊢ ( 𝑥 = 𝐴 → ( [ 𝑥 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) ) |
| 3 |
1 2
|
bitr3id |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) ) |