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Metamath Proof Explorer
Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014) Drop ax-12 . (Revised by GG, 1-Dec-2023)
|
|
Ref |
Expression |
|
Hypothesis |
sbcbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
sbcbidv |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑥 ] 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sbcbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
|
eqidd |
⊢ ( 𝜑 → 𝐴 = 𝐴 ) |
| 3 |
2 1
|
sbceqbid |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑥 ] 𝜒 ) ) |