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Mirrors > Home > ILE Home > Th. List > rexlimi | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexlimi.1 | ⊢ Ⅎ𝑥𝜓 |
rexlimi.2 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
rexlimi | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimi.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
2 | 1 | rgen 2374 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
3 | rexlimi.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | r19.23 2424 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
5 | 2, 4 | mpbi 133 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1349 ∈ wcel 1393 ∀wral 2306 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 df-rex 2312 |
This theorem is referenced by: rexlimiv 2427 r19.29af2 2452 triun 3867 reusv1 4190 reusv3 4192 onintrab2im 4244 fun11iun 5147 |
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