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Mirrors > Home > ILE Home > Th. List > exlimih | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
exlimih.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
exlimih.2 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
exlimih | ⊢ (∃𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimih.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | 19.23h 1474 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
3 | exlimih.2 | . 2 ⊢ (𝜑 → 𝜓) | |
4 | 2, 3 | mpgbi 1428 | 1 ⊢ (∃𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 ∃wex 1468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-gen 1425 ax-ie2 1470 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: exlimi 1573 exlimiv 1577 19.43 1607 hbex 1615 ax6blem 1628 19.41h 1663 ax9o 1676 equid 1677 equsex 1706 cbvexh 1728 equs5a 1766 sb5rf 1824 equvin 1835 euan 2055 moexexdc 2083 |
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