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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 4323. 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 4323 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | |
2 | 1 | ex 405 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
3 | noel 4184 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | pm2.21i 117 | . . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑) |
5 | 4 | rgen 3099 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
6 | raleq 3346 | . . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑)) | |
7 | 5, 6 | mpbiri 250 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
8 | 7 | necon3bi 2994 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
9 | exsimpl 1831 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴) | |
10 | df-rex 3095 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | n0 4197 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
12 | 9, 10, 11 | 3imtr4i 284 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
13 | 8, 12 | ja 175 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅) |
14 | 2, 13 | impbii 201 | 1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 ∅c0 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-dif 3833 df-nul 4180 |
This theorem is referenced by: iinpreima 6662 utopbas 22547 clsk3nimkb 39750 radcnvrat 40059 |
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