| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . . . . 6
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴) |
| 2 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 |
| 3 | 2 | a1i 11 |
. . . . . 6
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴) |
| 4 | | simp3 1139 |
. . . . . . 7
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 5 | | rabn0 4389 |
. . . . . . 7
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 6 | 4, 5 | sylibr 234 |
. . . . . 6
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅) |
| 7 | | nfrab1 3457 |
. . . . . . . 8
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
| 8 | 7 | nfcrii 2900 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
| 9 | 8 | bnj1228 35025 |
. . . . . 6
⊢ ((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) |
| 10 | 1, 3, 6, 9 | syl3anc 1373 |
. . . . 5
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) |
| 11 | | biid 261 |
. . . . 5
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)) |
| 12 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑥 𝑅 FrSe 𝐴 |
| 13 | | nfra1 3284 |
. . . . . . 7
⊢
Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) |
| 14 | | nfre1 3285 |
. . . . . . 7
⊢
Ⅎ𝑥∃𝑥 ∈ 𝐴 ¬ 𝜑 |
| 15 | 12, 13, 14 | nf3an 1901 |
. . . . . 6
⊢
Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 16 | 15 | nf5ri 2195 |
. . . . 5
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)) |
| 17 | 10, 11, 16 | bnj1521 34865 |
. . . 4
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)) |
| 18 | | eqid 2737 |
. . . . . 6
⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
| 19 | 18, 11 | bnj1212 34813 |
. . . . 5
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ 𝐴) |
| 20 | | nfra1 3284 |
. . . . . . . 8
⊢
Ⅎ𝑦∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 |
| 21 | | simp3 1139 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) |
| 22 | 21 | bnj1211 34811 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥)) |
| 23 | | con2b 359 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
| 24 | 23 | albii 1819 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
| 25 | 22, 24 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
| 26 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥) |
| 27 | | sp 2183 |
. . . . . . . . . . . . 13
⊢
(∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
| 28 | 25, 26, 27 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) |
| 29 | | simp1 1137 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦 ∈ 𝐴) |
| 30 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥𝐴 |
| 31 | 30 | elrabsf 3834 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑)) |
| 32 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑦 ∈ V |
| 33 | | sbcng 3836 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)) |
| 34 | 32, 33 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
| 35 | 34 | anbi2i 623 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑)) |
| 36 | 31, 35 | bitri 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑)) |
| 37 | 36 | notbii 320 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑)) |
| 38 | | imnan 399 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑)) |
| 39 | 37, 38 | sylbb2 238 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → (𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑)) |
| 40 | 39 | imp 406 |
. . . . . . . . . . . . 13
⊢ ((¬
𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑) |
| 41 | 40 | notnotrd 133 |
. . . . . . . . . . . 12
⊢ ((¬
𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → [𝑦 / 𝑥]𝜑) |
| 42 | 28, 29, 41 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑) |
| 43 | 42 | 3expa 1119 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑) |
| 44 | 43 | expcom 413 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)) |
| 45 | 44 | expd 415 |
. . . . . . . 8
⊢
(∀𝑦 ∈
{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))) |
| 46 | 20, 45 | ralrimi 3257 |
. . . . . . 7
⊢
(∀𝑦 ∈
{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)) |
| 47 | | bnj1204.1 |
. . . . . . 7
⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)) |
| 48 | 46, 47 | sylibr 234 |
. . . . . 6
⊢
(∀𝑦 ∈
{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → 𝜓) |
| 49 | 48 | 3ad2ant3 1136 |
. . . . 5
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓) |
| 50 | | simp12 1205 |
. . . . 5
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) |
| 51 | | simp3 1139 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) |
| 52 | 51 | bnj1211 34811 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑))) |
| 53 | | simp1 1137 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝑥 ∈ 𝐴) |
| 54 | | simp2 1138 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜓) |
| 55 | | sp 2183 |
. . . . . 6
⊢
(∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑))) |
| 56 | 52, 53, 54, 55 | syl3c 66 |
. . . . 5
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜑) |
| 57 | 19, 49, 50, 56 | syl3anc 1373 |
. . . 4
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑) |
| 58 | | rabid 3458 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
| 59 | 58 | simprbi 496 |
. . . . 5
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝜑) |
| 60 | 59 | 3ad2ant2 1135 |
. . . 4
⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑) |
| 61 | 17, 57, 60 | bnj1304 34833 |
. . 3
⊢ ¬
(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 62 | 61 | bnj1224 34815 |
. 2
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 63 | | dfral2 3099 |
. 2
⊢
(∀𝑥 ∈
𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
| 64 | 62, 63 | sylibr 234 |
1
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑) |