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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 117 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | alimi 1414 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
3 | exim 1561 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) |
5 | bi2 129 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
6 | 5 | alimi 1414 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑)) |
7 | exim 1561 | . . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑)) |
9 | 4, 8 | impbid 128 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1312 ∃wex 1451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-4 1470 ax-ial 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: exbii 1567 exbidh 1576 exintrbi 1595 19.19 1627 rexrnmpt 5517 |
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