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Metamath Proof Explorer
Description: Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011)
|
|
Ref |
Expression |
|
Hypothesis |
iotabidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
iotabidv |
⊢ ( 𝜑 → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iotabidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
1
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) ) |
| 3 |
|
iotabi |
⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) ) |
| 4 |
2 3
|
syl |
⊢ ( 𝜑 → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) ) |