P. 353 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bnj1023 35201	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜒)
Theorem	bnj1095 35202	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bnj1096 35203*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑))    ⇒   ⊢ (𝜓 → ∀𝑥𝜓)
Theorem	bnj1098 35204*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
Theorem	bnj1101 35205	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 → 𝜓)
Theorem	bnj1113 35206*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)
Theorem	bnj1109 35207	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥((𝐴 ≠ 𝐵 ∧ 𝜑) → 𝜓)    &   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
Theorem	bnj1131 35208	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	bnj1138 35209	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
Theorem	bnj1143 35210*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵
Theorem	bnj1146 35211*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵
Theorem	bnj1149 35212	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bnj1185 35213*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
Theorem	bnj1196 35214	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
Theorem	bnj1198 35215	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓′ ↔ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓′)
Theorem	bnj1209 35216*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))    ⇒   ⊢ (𝜒 → ∃𝑥𝜃)
Theorem	bnj1211 35217	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
Theorem	bnj1213 35218	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜃 → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜃 → 𝑥 ∈ 𝐵)
Theorem	bnj1212 35219*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))    ⇒   ⊢ (𝜃 → 𝑥 ∈ 𝐴)
Theorem	bnj1219 35220	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜁))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))    ⇒   ⊢ (𝜃 → 𝜓)
Theorem	bnj1224 35221	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ¬ (𝜃 ∧ 𝜏 ∧ 𝜂)    ⇒   ⊢ ((𝜃 ∧ 𝜏) → ¬ 𝜂)
Theorem	bnj1230 35222*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
Theorem	bnj1232 35223	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1235 35224	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	bnj1239 35225	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1238 35226	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1241 35227	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵)
Theorem	bnj1247 35228	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	bnj1254 35229	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	bnj1262 35230	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
Theorem	bnj1266 35231	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))    ⇒   ⊢ (𝜒 → ∃𝑥𝜓)
Theorem	bnj1265 35232*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1275 35233	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj1276 35234	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 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜃 → ∀𝑥𝜃)
Theorem	bnj1292 35235	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	bnj1293 35236	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	bnj1294 35237	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1299 35238	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1304 35239	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → ¬ 𝜒)    ⇒   ⊢ ¬ 𝜑
Theorem	bnj1316 35240*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	bnj1317 35241*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	bnj1322 35242	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
Theorem	bnj1340 35243	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∃𝑥𝜃)    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜓 → ∃𝑥𝜒)
Theorem	bnj1345 35244	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))    &   ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝜑 → ∃𝑥𝜃)
Theorem	bnj1350 35245*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj1351 35246*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
Theorem	bnj1352 35247*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
Theorem	bnj1361 35248*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	bnj1366 35249*	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 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))    ⇒   ⊢ (𝜓 → 𝐵 ∈ V)
Theorem	bnj1379 35250*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)    &   ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))    &   ⊢ (𝜒 ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑓 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓))    &   ⊢ (𝜏 ↔ (𝜃 ∧ 𝑔 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔))    ⇒   ⊢ (𝜓 → Fun ∪ 𝐴)
Theorem	bnj1383 35251*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)    &   ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))    ⇒   ⊢ (𝜓 → Fun ∪ 𝐴)
Theorem	bnj1385 35252*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)    &   ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))    &   ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)    &   ⊢ (𝜑′ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)    &   ⊢ 𝐸 = (dom ℎ ∩ dom 𝑔)    &   ⊢ (𝜓′ ↔ (𝜑′ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))    ⇒   ⊢ (𝜓 → Fun ∪ 𝐴)
Theorem	bnj1386 35253*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)    &   ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)    &   ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))    &   ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜓 → Fun ∪ 𝐴)
Theorem	bnj1397 35254	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1400 35255*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥
Theorem	bnj1405 35256*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)
Theorem	bnj1422 35257	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)    ⇒   ⊢ (𝜑 → 𝐴 Fn 𝐵)
Theorem	bnj1424 35258	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))
Theorem	bnj1436 35259	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 → 𝜑)
Theorem	bnj1441 35260*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2407. See bnj1441g 35261 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	bnj1441g 35261*	First-order logic and set theory. See bnj1441 35260 for a version with more disjoint variable conditions, but not requiring ax-13 2407. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	bnj1454 35262	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑))
Theorem	bnj1459 35263*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒)
Theorem	bnj1464 35264*	Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	bnj1465 35265*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)
Theorem	bnj1468 35266*	Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	bnj1476 35267	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}    &   ⊢ (𝜓 → 𝐷 = ∅)    ⇒   ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	bnj1502 35268	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	bnj1503 35269	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
Theorem	bnj1517 35270	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ 𝜓)}    ⇒   ⊢ (𝑥 ∈ 𝐴 → 𝜓)
Theorem	bnj1521 35271	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜒 → ∃𝑥𝜃)
Theorem	bnj1533 35272	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜃 → ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝐷)    &   ⊢ 𝐵 ⊆ 𝐴    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸}    ⇒   ⊢ (𝜃 → ∀𝑧 ∈ 𝐵 𝐶 = 𝐸)
Theorem	bnj1534 35273*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}    &   ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)    ⇒   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
Theorem	bnj1536 35274*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))
Theorem	bnj1538 35275	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 → 𝜑)
Theorem	bnj1541 35276	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝐴 ≠ 𝐵))    &   ⊢ ¬ 𝜑    ⇒   ⊢ (𝜓 → 𝐴 = 𝐵)
Theorem	bnj1542 35277*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐹 ≠ 𝐺)    &   ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥))
21.4.2  Well founded induction and recursion
Theorem	bnj110 35278*	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 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	bnj157 35279*	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 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 Fr 𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	bnj66 35280*	Technical lemma for bnj60 35482. 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 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)
Theorem	bnj91 35281*	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))
Theorem	bnj92 35282*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
Theorem	bnj93 35283*	Technical lemma for bnj97 35286. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Theorem	bnj95 35284	Technical lemma for bnj124 35291. 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
Theorem	bnj96 35285*	Technical lemma for bnj150 35296. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)
Theorem	bnj97 35286*	Technical lemma for bnj150 35296. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Theorem	bnj98 35287	Technical lemma for bnj150 35296. 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 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
Theorem	bnj106 35288*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
Theorem	bnj118 35289*	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))
Theorem	bnj121 35290*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)    &   ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)    &   ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)    ⇒   ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
Theorem	bnj124 35291*	Technical lemma for bnj150 35296. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))    ⇒   ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))
Theorem	bnj125 35292*	Technical lemma for bnj150 35296. 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))
Theorem	bnj126 35293*	Technical lemma for bnj150 35296. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) 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 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
Theorem	bnj130 35294*	Technical lemma for bnj151 35297. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)    &   ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)    &   ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)    ⇒   ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
Theorem	bnj149 35295*	Technical lemma for bnj151 35297. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))    &   ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))    &   ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)    &   ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)    &   ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)    &   ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    ⇒   ⊢ 𝜃1
Theorem	bnj150 35296*	Technical lemma for bnj151 35297. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)    &   ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)    &   ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))    &   ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)    &   ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)    &   ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)    &   ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)    ⇒   ⊢ 𝜃0
Theorem	bnj151 35297*	Technical lemma for bnj153 35300. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))    &   ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)    &   ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)    &   ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)    &   ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))    &   ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))    &   ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)    &   ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)    &   ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)    &   ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)    &   ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))    &   ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)    &   ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)    &   ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)    ⇒   ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))
Theorem	bnj154 35298*	Technical lemma for bnj153 35300. 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.)
⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)    &   ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    ⇒   ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
Theorem	bnj155 35299*	Technical lemma for bnj153 35300. 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.)
⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)    &   ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    ⇒   ⊢ (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
Theorem	bnj153 35300*	Technical lemma for bnj852 35341. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ⊢ 𝐷 = (ω ∖ {∅})    &   ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))    &   ⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))    ⇒   ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))