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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1185 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1185.1 | ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧) |
Ref | Expression |
---|---|
bnj1185 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1185.1 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧) | |
2 | breq1 5151 | . . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑧 ↔ 𝑦𝑅𝑧)) | |
3 | 2 | notbid 318 | . . . . 5 ⊢ (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧)) |
4 | 3 | cbvralvw 3235 | . . . 4 ⊢ (∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) |
5 | 4 | rexbii 3092 | . . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) |
6 | 1, 5 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) |
7 | eleq1w 2822 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
8 | breq2 5152 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥)) | |
9 | 8 | notbid 318 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥)) |
10 | 9 | ralbidv 3176 | . . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) |
11 | 7, 10 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))) |
12 | 11 | cbvexvw 2034 | . . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) |
13 | df-rex 3069 | . . 3 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧)) | |
14 | df-rex 3069 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) | |
15 | 12, 13, 14 | 3bitr4ri 304 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑧) |
16 | 6, 15 | sylibr 234 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: bnj1190 35001 bnj1189 35002 |
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