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Mirrors > Home > ILE Home > Th. List > r19.36av | GIF version |
Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if 𝐴 has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
r19.36av | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.35-1 2460 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | idd 21 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → 𝜓)) | |
3 | 2 | rexlimiv 2427 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓) |
4 | 3 | imim2i 12 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
5 | 1, 4 | syl 14 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ 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-17 1419 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: iinss 3708 |
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