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Mirrors > Home > MPE Home > Th. List > r19.35 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.35 1875. (Contributed by NM, 20-Sep-2003.) (Proof shortened by Wolf Lammen, 22-Dec-2024.) |
Ref | Expression |
---|---|
r19.35 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.5 361 | . . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓)) | |
2 | 1 | ralrexbid 3104 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
3 | 2 | biimpcd 249 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
4 | rexnal 3098 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑) | |
5 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 5 | reximi 3082 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
7 | 4, 6 | sylbir 235 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
8 | ax-1 6 | . . . 4 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
9 | 8 | reximi 3082 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
10 | 7, 9 | ja 186 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
11 | 3, 10 | impbii 209 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wral 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-ral 3060 df-rex 3069 |
This theorem is referenced by: r19.43 3120 r19.37v 3180 r19.36v 3182 r19.37 3260 r19.37zv 4508 r19.36zv 4513 iinexg 5354 bndndx 12523 nmobndseqi 30808 nmobndseqiALT 30809 unielss 43207 r19.36vf 45076 |
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