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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 1881. (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 | pm5.5 362 | . . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓)) | |
2 | 1 | ralrexbid 3107 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
3 | 2 | biimpcd 248 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
4 | rexnal 3101 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑) | |
5 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 5 | reximi 3085 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
7 | 4, 6 | sylbir 234 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
8 | ax-1 6 | . . . 4 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
9 | 8 | reximi 3085 | . . 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 208 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wral 3062 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-ral 3063 df-rex 3072 |
This theorem is referenced by: r19.43 3123 r19.37v 3182 r19.36v 3184 r19.37 3260 r19.37zv 4502 r19.36zv 4507 iinexg 5342 bndndx 12471 nmobndseqi 30032 nmobndseqiALT 30033 unielss 41967 r19.36vf 43825 |
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