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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj206 31901 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′[𝑀 / 𝑛]𝜑)    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   (𝜒′[𝑀 / 𝑛]𝜒)    &   𝑀 ∈ V       ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
 
Theorembnj216 31902 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴 = suc 𝐵𝐵𝐴)
 
Theorembnj219 31903 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑛 = suc 𝑚𝑚 E 𝑛)
 
Theorembnj226 31904* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵𝐶        𝑥𝐴 𝐵𝐶
 
Theorembnj228 31905 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       ((𝑥𝐴𝜑) → 𝜓)
 
Theorembnj519 31906 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
𝐴 ∈ V       (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})
 
Theorembnj521 31907 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 ∩ {𝐴}) = ∅
 
Theorembnj524 31908 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜓)    &   𝐴 ∈ V       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)
 
Theorembnj525 31909* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 ∈ V       ([𝐴 / 𝑥]𝜑𝜑)
 
Theorembnj534 31910* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → (∃𝑥𝜑𝜓))       (𝜒 → ∃𝑥(𝜑𝜓))
 
Theorembnj538 31911* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.)
𝐴 ∈ V       ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
 
Theorembnj529 31912 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑀𝐷 → ∅ ∈ 𝑀)
 
Theorembnj551 31913 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
 
Theorembnj563 31914 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))       ((𝜂𝜌) → suc 𝑖𝑚)
 
Theorembnj564 31915 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))       (𝜏 → dom 𝑓 = 𝑚)
 
Theorembnj593 31916 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∃𝑥𝜒)
 
Theorembnj596 31917 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∃𝑥𝜓)       (𝜑 → ∃𝑥(𝜑𝜓))
 
Theorembnj610 31918* Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable restriction ($d 𝐴 𝑥). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜓′))    &   (𝑦 = 𝐴 → (𝜓′𝜓))       ([𝐴 / 𝑥]𝜑𝜓)
 
Theorembnj642 31919 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜑)
 
Theorembnj643 31920 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜓)
 
Theorembnj645 31921 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜃)
 
Theorembnj658 31922 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜑𝜓𝜒))
 
Theorembnj667 31923 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorembnj705 31924 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj706 31925 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj707 31926 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj708 31927 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj721 31928 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj832 31929 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj835 31930 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj836 31931 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜓𝜏)       (𝜂𝜏)
 
Theorembnj837 31932 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜒𝜏)       (𝜂𝜏)
 
Theorembnj769 31933 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj770 31934 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜓𝜏)       (𝜂𝜏)
 
Theorembnj771 31935 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜒𝜏)       (𝜂𝜏)
 
Theorembnj887 31936 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜑′)    &   (𝜓𝜓′)    &   (𝜒𝜒′)    &   (𝜃𝜃′)       ((𝜑𝜓𝜒𝜃) ↔ (𝜑′𝜓′𝜒′𝜃′))
 
Theorembnj918 31937 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       𝐺 ∈ V
 
Theorembnj919 31938* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝐹 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑃 / 𝑛]𝜑)    &   (𝜓′[𝑃 / 𝑛]𝜓)    &   (𝜒′[𝑃 / 𝑛]𝜒)    &   𝑃 ∈ V       (𝜒′ ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′))
 
Theorembnj923 31939 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑛𝐷𝑛 ∈ ω)
 
Theorembnj927 31940 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 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
 
Theorembnj930 31941 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐹 Fn 𝐴)       (𝜑 → Fun 𝐹)
 
Theorembnj931 31942 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐵𝐴
 
Theorembnj937 31943* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)       (𝜑𝜓)
 
Theorembnj941 31944 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 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
 
Theorembnj945 31945 Technical lemma for bnj69 32180. 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 𝑛𝑝 = suc 𝑛𝐴𝑛) → (𝐺𝐴) = (𝑓𝐴))
 
Theorembnj946 31946 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       (𝜑 ↔ ∀𝑥(𝑥𝐴𝜓))
 
Theorembnj951 31947 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜏𝜑)    &   (𝜏𝜓)    &   (𝜏𝜒)    &   (𝜏𝜃)       (𝜏 → (𝜑𝜓𝜒𝜃))
 
Theorembnj956 31948 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 31949* 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 31950 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 31951* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
 
Theorembnj1023 31952 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜓𝜒)       𝑥(𝜑𝜒)
 
Theorembnj1095 31953 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       (𝜑 → ∀𝑥𝜑)
 
Theorembnj1096 31954* 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 31955* 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 31956 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜒𝜑)       𝑥(𝜒𝜓)
 
Theorembnj1113 31957* 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 31958 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 31959 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   𝑥𝜑       𝜑
 
Theorembnj1138 31960 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
 
Theorembnj1142 31961 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 31962* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝐴 𝐵𝐵
 
Theorembnj1146 31963* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)        𝑥𝐴 𝐵𝐵
 
Theorembnj1149 31964 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 31965* 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 31966 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 31967 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 31968* 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 31969 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 31970 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜃𝑥𝐴)       (𝜃𝑥𝐵)
 
Theorembnj1212 31971* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}    &   (𝜃 ↔ (𝜒𝑥𝐵𝜏))       (𝜃𝑥𝐴)
 
Theorembnj1219 31972 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝜑𝜓𝜁))    &   (𝜃 ↔ (𝜒𝜏𝜂))       (𝜃𝜓)
 
Theorembnj1224 31973 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
¬ (𝜃𝜏𝜂)       ((𝜃𝜏) → ¬ 𝜂)
 
Theorembnj1230 31974* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}       (𝑦𝐵 → ∀𝑥 𝑦𝐵)
 
Theorembnj1232 31975 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜓)
 
Theorembnj1235 31976 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜒)
 
Theorembnj1239 31977 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜓𝜒) → ∃𝑥𝐴 𝜓)
 
Theorembnj1238 31978 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 31979 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐴𝐵)    &   (𝜓𝐶 = 𝐴)       ((𝜑𝜓) → 𝐶𝐵)
 
Theorembnj1247 31980 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜃)
 
Theorembnj1254 31981 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜏)
 
Theorembnj1262 31982 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜑𝐶 = 𝐴)       (𝜑𝐶𝐵)
 
Theorembnj1266 31983 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥(𝜑𝜓))       (𝜒 → ∃𝑥𝜓)
 
Theorembnj1265 31984* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜓)
 
Theorembnj1275 31985 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 31986 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 31987 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐵
 
Theorembnj1293 31988 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐶
 
Theorembnj1294 31989 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝜓)
 
Theorembnj1299 31990 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 31991 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 31992* 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 31993* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theorembnj1322 31994 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
 
Theorembnj1340 31995 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 31996 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 31997* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
 
Theorembnj1351 31998* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theorembnj1352 31999* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theorembnj1361 32000* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
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