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| Mirrors > Home > MPE Home > Th. List > r19.2zb | Structured version Visualization version GIF version | ||
| Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4495. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.) |
| Ref | Expression |
|---|---|
| r19.2zb | ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.2z 4495 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | |
| 2 | 1 | ex 412 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
| 3 | noel 4338 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
| 4 | 3 | pm2.21i 119 | . . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑) |
| 5 | 4 | rgen 3063 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
| 6 | raleq 3323 | . . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑)) | |
| 7 | 5, 6 | mpbiri 258 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
| 8 | 7 | necon3bi 2967 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
| 9 | exsimpl 1868 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴) | |
| 10 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 11 | n0 4353 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 12 | 9, 10, 11 | 3imtr4i 292 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
| 13 | 8, 12 | ja 186 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅) |
| 14 | 2, 13 | impbii 209 | 1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: iinpreima 7089 utopbas 24244 clsk3nimkb 44053 radcnvrat 44333 |
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