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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj23 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj23.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Ref | Expression |
---|---|
bnj23 | ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcng 3816 | . . . . 5 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑)) | |
2 | 1 | elv 3497 | . . . 4 ⊢ ([𝑤 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑤 / 𝑥]𝜑) |
3 | bnj23.1 | . . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | |
4 | 3 | eleq2i 2901 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
5 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
6 | 5 | elrabsf 3813 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑)) |
7 | 4, 6 | bitri 276 | . . . . . 6 ⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑)) |
8 | breq1 5060 | . . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑤𝑅𝑦)) | |
9 | 8 | notbid 319 | . . . . . . 7 ⊢ (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦)) |
10 | 9 | rspccv 3617 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑦)) |
11 | 7, 10 | syl5bir 244 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥] ¬ 𝜑) → ¬ 𝑤𝑅𝑦)) |
12 | 11 | expdimp 453 | . . . 4 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → ([𝑤 / 𝑥] ¬ 𝜑 → ¬ 𝑤𝑅𝑦)) |
13 | 2, 12 | syl5bir 244 | . . 3 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (¬ [𝑤 / 𝑥]𝜑 → ¬ 𝑤𝑅𝑦)) |
14 | 13 | con4d 115 | . 2 ⊢ ((∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 ∧ 𝑤 ∈ 𝐴) → (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑)) |
15 | 14 | ralrimiva 3179 | 1 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 Vcvv 3492 [wsbc 3769 class class class wbr 5057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 |
This theorem is referenced by: bnj110 32029 |
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