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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj956 30101 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)       (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theorembnj976 30102* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))    &   (𝜑′[𝐺 / 𝑓]𝜑)    &   (𝜓′[𝐺 / 𝑓]𝜓)    &   (𝜒′[𝐺 / 𝑓]𝜒)    &   𝐺 ∈ V       (𝜒′ ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))

Theorembnj982 30103 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜃 → ∀𝑥𝜃)       ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))

Theorembnj1019 30104* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))

Theorembnj1023 30105 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜓𝜒)       𝑥(𝜑𝜒)

Theorembnj1095 30106 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       (𝜑 → ∀𝑥𝜑)

Theorembnj1096 30107* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 ↔ (𝜒𝜃𝜏𝜑))       (𝜓 → ∀𝑥𝜓)

Theorembnj1098 30108* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))

Theorembnj1101 30109 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜒𝜑)       𝑥(𝜒𝜓)

Theorembnj1113 30110* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴 = 𝐵 𝑥𝐶 𝐸 = 𝑥𝐷 𝐸)

Theorembnj1109 30111 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥((𝐴𝐵𝜑) → 𝜓)    &   ((𝐴 = 𝐵𝜑) → 𝜓)       𝑥(𝜑𝜓)

Theorembnj1131 30112 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   𝑥𝜑       𝜑

Theorembnj1138 30113 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))

Theorembnj1142 30114 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥(𝑥𝐴𝜓))       (𝜑 → ∀𝑥𝐴 𝜓)

Theorembnj1143 30115* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝐴 𝐵𝐵

Theorembnj1146 30116* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)        𝑥𝐴 𝐵𝐵

Theorembnj1149 30117 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)       (𝜑 → (𝐴𝐵) ∈ V)

Theorembnj1185 30118* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧)       (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theorembnj1196 30119 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥(𝑥𝐴𝜓))

Theorembnj1198 30120 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓′𝜓)       (𝜑 → ∃𝑥𝜓′)

Theorembnj1209 30121* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥𝐵 𝜑)    &   (𝜃 ↔ (𝜒𝑥𝐵𝜑))       (𝜒 → ∃𝑥𝜃)

Theorembnj1211 30122 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝐴 𝜓)       (𝜑 → ∀𝑥(𝑥𝐴𝜓))

Theorembnj1213 30123 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜃𝑥𝐴)       (𝜃𝑥𝐵)

Theorembnj1212 30124* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}    &   (𝜃 ↔ (𝜒𝑥𝐵𝜏))       (𝜃𝑥𝐴)

Theorembnj1219 30125 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝜑𝜓𝜁))    &   (𝜃 ↔ (𝜒𝜏𝜂))       (𝜃𝜓)

Theorembnj1224 30126 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
¬ (𝜃𝜏𝜂)       ((𝜃𝜏) → ¬ 𝜂)

Theorembnj1230 30127* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}       (𝑦𝐵 → ∀𝑥 𝑦𝐵)

Theorembnj1232 30128 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜓)

Theorembnj1235 30129 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜒)

Theorembnj1239 30130 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜓𝜒) → ∃𝑥𝐴 𝜓)

Theorembnj1238 30131 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥𝐴 (𝜓𝜒))       (𝜑 → ∃𝑥𝐴 𝜓)

Theorembnj1241 30132 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐴𝐵)    &   (𝜓𝐶 = 𝐴)       ((𝜑𝜓) → 𝐶𝐵)

Theorembnj1247 30133 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜃)

Theorembnj1254 30134 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜏)

Theorembnj1262 30135 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜑𝐶 = 𝐴)       (𝜑𝐶𝐵)

Theorembnj1266 30136 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥(𝜑𝜓))       (𝜒 → ∃𝑥𝜓)

Theorembnj1265 30137* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜓)

Theorembnj1275 30138 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥(𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥(𝜑𝜓𝜒))

Theorembnj1276 30139 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜃 ↔ (𝜑𝜓𝜒))       (𝜃 → ∀𝑥𝜃)

Theorembnj1292 30140 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐵

Theorembnj1293 30141 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐶

Theorembnj1294 30142 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝜓)

Theorembnj1299 30143 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 (𝜓𝜒))       (𝜑 → ∃𝑥𝐴 𝜓)

Theorembnj1304 30144 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓𝜒)    &   (𝜓 → ¬ 𝜒)        ¬ 𝜑

Theorembnj1316 30145* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theorembnj1317 30146* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theorembnj1322 30147 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)

Theorembnj1340 30148 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∃𝑥𝜃)    &   (𝜒 ↔ (𝜓𝜃))    &   (𝜓 → ∀𝑥𝜓)       (𝜓 → ∃𝑥𝜒)

Theorembnj1345 30149 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥(𝜓𝜒))    &   (𝜃 ↔ (𝜑𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥𝜃)

Theorembnj1350 30150* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theorembnj1351 30151* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorembnj1352 30152* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorembnj1361 30153* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theorembnj1366 30154* 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.)
(𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))       (𝜓𝐵 ∈ V)

Theorembnj1379 30155* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝜒 ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))    &   (𝜃 ↔ (𝜒𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓))    &   (𝜏 ↔ (𝜃𝑔𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔))       (𝜓 → Fun 𝐴)

Theorembnj1383 30156* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))       (𝜓 → Fun 𝐴)

Theorembnj1385 30157* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝑥𝐴 → ∀𝑓 𝑥𝐴)    &   (𝜑′ ↔ ∀𝐴 Fun )    &   𝐸 = (dom ∩ dom 𝑔)    &   (𝜓′ ↔ (𝜑′ ∧ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸)))       (𝜓 → Fun 𝐴)

Theorembnj1386 30158* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)    &   𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))    &   (𝑥𝐴 → ∀𝑓 𝑥𝐴)       (𝜓 → Fun 𝐴)

Theorembnj1397 30159 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑𝜓)

Theorembnj1400 30160* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       dom 𝐴 = 𝑥𝐴 dom 𝑥

Theorembnj1405 30161* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝑋 𝑦𝐴 𝐵)       (𝜑 → ∃𝑦𝐴 𝑋𝐵)

Theorembnj1422 30162 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐴)    &   (𝜑 → dom 𝐴 = 𝐵)       (𝜑𝐴 Fn 𝐵)

Theorembnj1424 30163 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝐷𝐴 → (𝐷𝐵𝐷𝐶))

Theorembnj1436 30164 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theorembnj1441 30165* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑥𝐴 → ∀𝑦 𝑥𝐴)    &   (𝜑 → ∀𝑦𝜑)       (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})

Theorembnj1454 30166 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))

Theorembnj1459 30167* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ (𝜑𝑥𝐴))    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝐴 𝜒)

Theorembnj1464 30168* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theorembnj1465 30169* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒𝜓)       ((𝜒𝐴𝑉) → ∃𝑥𝜑)

Theorembnj1468 30170* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theorembnj1476 30171 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = {𝑥𝐴 ∣ ¬ 𝜑}    &   (𝜓𝐷 = ∅)       (𝜓 → ∀𝑥𝐴 𝜑)

Theorembnj1502 30172 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴 ∈ dom 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembnj1503 30173 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴 ⊆ dom 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembnj1517 30174 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥 ∣ (𝜑𝜓)}       (𝑥𝐴𝜓)

Theorembnj1521 30175 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥𝐵 𝜑)    &   (𝜃 ↔ (𝜒𝑥𝐵𝜑))    &   (𝜒 → ∀𝑥𝜒)       (𝜒 → ∃𝑥𝜃)

Theorembnj1533 30176 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 → ∀𝑧𝐵 ¬ 𝑧𝐷)    &   𝐵𝐴    &   𝐷 = {𝑧𝐴𝐶𝐸}       (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)

Theorembnj1534 30177* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}

Theorembnj1536 30178* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (𝐹𝐵) = (𝐺𝐵))

Theorembnj1538 30179 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴𝜑)

Theorembnj1541 30180 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝐴𝐵))    &    ¬ 𝜑       (𝜓𝐴 = 𝐵)

Theorembnj1542 30181* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐹𝐺)    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))

21.4.2  Well founded induction and recursion

Theorembnj110 30182* Well-founded induction restricted to a set (𝐴 ∈ V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 ∈ V    &   (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))       ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)

Theorembnj157 30183* Well-founded induction restricted to a set (𝐴 ∈ V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))    &   𝐴 ∈ V    &   𝑅 Fr 𝐴       (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 𝜑)

Theorembnj66 30184* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑔𝐶 → Rel 𝑔)

Theorembnj91 30185* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   𝑍 ∈ V       ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj92 30186* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝑍 ∈ V       ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj93 30187* Technical lemma for bnj97 30190. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)

Theorembnj95 30188 Technical lemma for bnj124 30195. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       𝐹 ∈ V

Theorembnj96 30189* Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)

Theorembnj97 30190* Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj98 30191 Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))

Theorembnj106 30192* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐹 ∈ V       ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj118 30193* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜑′[1𝑜 / 𝑛]𝜑)       (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj121 30194* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜁′[1𝑜 / 𝑛]𝜁)    &   (𝜑′[1𝑜 / 𝑛]𝜑)    &   (𝜓′[1𝑜 / 𝑛]𝜓)       (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))

Theorembnj124 30195* Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)    &   (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))       (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))

Theorembnj125 30196* Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜑′[1𝑜 / 𝑛]𝜑)    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj126 30197* Technical lemma for bnj150 30200. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[1𝑜 / 𝑛]𝜓)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj130 30198* Technical lemma for bnj151 30201. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1𝑜 / 𝑛]𝜑)    &   (𝜓′[1𝑜 / 𝑛]𝜓)    &   (𝜃′[1𝑜 / 𝑛]𝜃)       (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))

Theorembnj149 30199* Technical lemma for bnj151 30201. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))    &   (𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))       𝜃1

Theorembnj150 30200* Technical lemma for bnj151 30201. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1𝑜 / 𝑛]𝜑)    &   (𝜓′[1𝑜 / 𝑛]𝜓)    &   (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))    &   (𝜁′[1𝑜 / 𝑛]𝜁)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)       𝜃0

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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