![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > exbi | GIF version |
Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
exbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi1 116 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | alimi 1389 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
3 | exim 1535 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) |
5 | bi2 128 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
6 | 5 | alimi 1389 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑)) |
7 | exim 1535 | . . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑)) |
9 | 4, 8 | impbid 127 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1287 ∃wex 1426 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-4 1445 ax-ial 1472 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: exbii 1541 exbidh 1550 exintrbi 1569 19.19 1601 rexrnmpt 5442 |
Copyright terms: Public domain | W3C validator |