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Metamath Proof Explorer
Description: Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005)
|
|
Ref |
Expression |
|
Hypothesis |
sbcbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
sbcbii |
⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sbcbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
| 2 |
1
|
a1i |
⊢ ( ⊤ → ( 𝜑 ↔ 𝜓 ) ) |
| 3 |
2
|
sbcbidv |
⊢ ( ⊤ → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) |
| 4 |
3
|
mptru |
⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) |