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Metamath Proof Explorer
Description: Theorem 19.19 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)
|
|
Ref |
Expression |
|
Hypothesis |
19.19.1 |
⊢ Ⅎ 𝑥 𝜑 |
|
Assertion |
19.19 |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∃ 𝑥 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
19.19.1 |
⊢ Ⅎ 𝑥 𝜑 |
| 2 |
1
|
19.9 |
⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 ) |
| 3 |
|
exbi |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) ) |
| 4 |
2 3
|
bitr3id |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∃ 𝑥 𝜓 ) ) |