P. 350 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34901-35000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bnj832 34901	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj835 34902	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj836 34903	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj837 34904	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj769 34905	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj770 34906	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj771 34907	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj887 34908	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜑′)    &   ⊢ (𝜓 ↔ 𝜓′)    &   ⊢ (𝜒 ↔ 𝜒′)    &   ⊢ (𝜃 ↔ 𝜃′)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′ ∧ 𝜃′))
Theorem	bnj918 34909	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    ⇒   ⊢ 𝐺 ∈ V
Theorem	bnj919 34910*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜑′ ↔ [𝑃 / 𝑛]𝜑)    &   ⊢ (𝜓′ ↔ [𝑃 / 𝑛]𝜓)    &   ⊢ (𝜒′ ↔ [𝑃 / 𝑛]𝜒)    &   ⊢ 𝑃 ∈ V    ⇒   ⊢ (𝜒′ ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))
Theorem	bnj923 34911	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
Theorem	bnj927 34912	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
Theorem	bnj931 34913	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	bnj937 34914*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj941 34915	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    ⇒   ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
Theorem	bnj945 34916	Technical lemma for bnj69 35152. 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.)
⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    ⇒   ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (𝐺‘𝐴) = (𝑓‘𝐴))
Theorem	bnj946 34917	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	bnj951 34918	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜏 → 𝜑)    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝜏 → 𝜒)    &   ⊢ (𝜏 → 𝜃)    ⇒   ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	bnj956 34919	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 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	bnj976 34920*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓))    &   ⊢ (𝜑′ ↔ [𝐺 / 𝑓]𝜑)    &   ⊢ (𝜓′ ↔ [𝐺 / 𝑓]𝜓)    &   ⊢ (𝜒′ ↔ [𝐺 / 𝑓]𝜒)    &   ⊢ 𝐺 ∈ V    ⇒   ⊢ (𝜒′ ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′))
Theorem	bnj982 34921	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)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	bnj1019 34922*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))
Theorem	bnj1023 34923	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 34924	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 34925*	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 34926*	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 34927	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 34928*	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 34929	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 34930	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	bnj1138 34931	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 34932*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵
Theorem	bnj1146 34933*	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 34934	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 34935*	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 34936	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 34937	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 34938*	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 34939	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 34940	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 34941*	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 34942	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 34943	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 34944*	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 34945	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 34946	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 34947	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 34948	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 34949	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 34950	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 34951	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 34952	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 34953	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 34954*	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 34955	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 34956	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 34957	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 34958	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 34959	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 34960	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 34961	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 34962*	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 34963*	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 34964	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 34965	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 34966	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 34967*	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 34968*	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 34969*	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 34970*	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 34971*	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 34972*	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 34973*	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 34974*	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 34975*	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 34976	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 34977*	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 34978*	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 34979	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 34980	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 34981	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 34982*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2376. See bnj1441g 34983 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 34983*	First-order logic and set theory. See bnj1441 34982 for a version with more disjoint variable conditions, but not requiring ax-13 2376. (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 34984	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 34985*	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 34986*	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 34987*	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 34988*	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 34989	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 34990	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 34991	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 34992	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 34993	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 34994	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 34995*	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 34996*	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 34997	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 34998	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 34999*	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 35000*	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)