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Metamath Proof Explorer
Description: One direction of sbor , using fewer axioms. Compare 19.33 .
(Contributed by Steven Nguyen, 18-Aug-2023)
|
|
Ref |
Expression |
|
Assertion |
sbor2 |
⊢ ( ( [ 𝑡 / 𝑥 ] 𝜑 ∨ [ 𝑡 / 𝑥 ] 𝜓 ) → [ 𝑡 / 𝑥 ] ( 𝜑 ∨ 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
orc |
⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) ) |
| 2 |
1
|
sbimi |
⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] ( 𝜑 ∨ 𝜓 ) ) |
| 3 |
|
olc |
⊢ ( 𝜓 → ( 𝜑 ∨ 𝜓 ) ) |
| 4 |
3
|
sbimi |
⊢ ( [ 𝑡 / 𝑥 ] 𝜓 → [ 𝑡 / 𝑥 ] ( 𝜑 ∨ 𝜓 ) ) |
| 5 |
2 4
|
jaoi |
⊢ ( ( [ 𝑡 / 𝑥 ] 𝜑 ∨ [ 𝑡 / 𝑥 ] 𝜓 ) → [ 𝑡 / 𝑥 ] ( 𝜑 ∨ 𝜓 ) ) |