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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 1885. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
r19.35 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 42 | . . . . 5 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | |
2 | 1 | ralimi 3087 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓)) |
3 | rexim 3169 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓) → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)) |
5 | 4 | com12 32 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
6 | rexnal 3166 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑) | |
7 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
8 | 7 | reximi 3175 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
9 | 6, 8 | sylbir 238 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
10 | ax-1 6 | . . . 4 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
11 | 10 | reximi 3175 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
12 | 9, 11 | ja 189 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
13 | 5, 12 | impbii 212 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wral 3064 ∃wrex 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-ral 3069 df-rex 3070 |
This theorem is referenced by: r19.36v 3271 r19.37 3272 r19.37v 3273 r19.43 3279 r19.37zv 4430 r19.36zv 4435 iinexg 5251 bndndx 12119 nmobndseqi 28892 nmobndseqiALT 28893 r19.36vf 42406 |
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