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Metamath Proof Explorer
Description: Closed form of 19.41 from the same axioms as 19.41v . The same is
doable with 19.27 , 19.28 , 19.31 , 19.32 , 19.44 , 19.45 .
(Contributed by BJ, 2-Dec-2023)
|
|
Ref |
Expression |
|
Assertion |
bj-19.41t |
⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
exancom |
⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) ) |
| 2 |
|
bj-19.42t |
⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ∃ 𝑥 𝜑 ) ) ) |
| 3 |
1 2
|
bitrid |
⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ ∃ 𝑥 𝜑 ) ) ) |
| 4 |
3
|
biancomd |
⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) ) |