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Theorem List for Metamath Proof Explorer - 32001-32100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj206 32001 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 32002 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴 = suc 𝐵𝐵𝐴)

Theorembnj219 32003 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑛 = suc 𝑚𝑚 E 𝑛)

Theorembnj226 32004* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵𝐶        𝑥𝐴 𝐵𝐶

Theorembnj228 32005 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 32006 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 32007 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 ∩ {𝐴}) = ∅

Theorembnj524 32008 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 32009* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 ∈ V       ([𝐴 / 𝑥]𝜑𝜑)

Theorembnj534 32010* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → (∃𝑥𝜑𝜓))       (𝜒 → ∃𝑥(𝜑𝜓))

Theorembnj538 32011* 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 32012 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑀𝐷 → ∅ ∈ 𝑀)

Theorembnj551 32013 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 32014 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 32015 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 32016 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∃𝑥𝜒)

Theorembnj596 32017 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 32018* 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 32019 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜑)

Theorembnj643 32020 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜓)

Theorembnj645 32021 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜃)

Theorembnj658 32022 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜑𝜓𝜒))

Theorembnj667 32023 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜓𝜒𝜃))

Theorembnj705 32024 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)

Theorembnj706 32025 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)

Theorembnj707 32026 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)

Theorembnj708 32027 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)

Theorembnj721 32028 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)

Theorembnj832 32029 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓))    &   (𝜑𝜏)       (𝜂𝜏)

Theorembnj835 32030 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜑𝜏)       (𝜂𝜏)

Theorembnj836 32031 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜓𝜏)       (𝜂𝜏)

Theorembnj837 32032 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜒𝜏)       (𝜂𝜏)

Theorembnj769 32033 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜑𝜏)       (𝜂𝜏)

Theorembnj770 32034 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜓𝜏)       (𝜂𝜏)

Theorembnj771 32035 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜒𝜏)       (𝜂𝜏)

Theorembnj887 32036 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜑′)    &   (𝜓𝜓′)    &   (𝜒𝜒′)    &   (𝜃𝜃′)       ((𝜑𝜓𝜒𝜃) ↔ (𝜑′𝜓′𝜒′𝜃′))

Theorembnj918 32037 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 32038* 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 32039 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑛𝐷𝑛 ∈ ω)

Theorembnj927 32040 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 32041 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 32042 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐵𝐴

Theorembnj937 32043* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)       (𝜑𝜓)

Theorembnj941 32044 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 32045 Technical lemma for bnj69 32282. 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 32046 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 32047 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜏𝜑)    &   (𝜏𝜓)    &   (𝜏𝜒)    &   (𝜏𝜃)       (𝜏 → (𝜑𝜓𝜒𝜃))

Theorembnj956 32048 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 32049* 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 32050 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 32051* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))

Theorembnj1023 32052 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜓𝜒)       𝑥(𝜑𝜒)

Theorembnj1095 32053 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       (𝜑 → ∀𝑥𝜑)

Theorembnj1096 32054* 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 32055* 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 32056 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜒𝜑)       𝑥(𝜒𝜓)

Theorembnj1113 32057* 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 32058 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 32059 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   𝑥𝜑       𝜑

Theorembnj1138 32060 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))

Theorembnj1142 32061 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 32062* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝐴 𝐵𝐵

Theorembnj1146 32063* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)        𝑥𝐴 𝐵𝐵

Theorembnj1149 32064 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 32065* 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 32066 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 32067 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 32068* 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 32069 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 32070 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜃𝑥𝐴)       (𝜃𝑥𝐵)

Theorembnj1212 32071* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}    &   (𝜃 ↔ (𝜒𝑥𝐵𝜏))       (𝜃𝑥𝐴)

Theorembnj1219 32072 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝜑𝜓𝜁))    &   (𝜃 ↔ (𝜒𝜏𝜂))       (𝜃𝜓)

Theorembnj1224 32073 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
¬ (𝜃𝜏𝜂)       ((𝜃𝜏) → ¬ 𝜂)

Theorembnj1230 32074* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}       (𝑦𝐵 → ∀𝑥 𝑦𝐵)

Theorembnj1232 32075 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜓)

Theorembnj1235 32076 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜒)

Theorembnj1239 32077 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜓𝜒) → ∃𝑥𝐴 𝜓)

Theorembnj1238 32078 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 32079 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐴𝐵)    &   (𝜓𝐶 = 𝐴)       ((𝜑𝜓) → 𝐶𝐵)

Theorembnj1247 32080 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜃)

Theorembnj1254 32081 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜏)

Theorembnj1262 32082 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜑𝐶 = 𝐴)       (𝜑𝐶𝐵)

Theorembnj1266 32083 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥(𝜑𝜓))       (𝜒 → ∃𝑥𝜓)

Theorembnj1265 32084* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜓)

Theorembnj1275 32085 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 32086 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 32087 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐵

Theorembnj1293 32088 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐶

Theorembnj1294 32089 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝜓)

Theorembnj1299 32090 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 32091 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 32092* 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 32093* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theorembnj1322 32094 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)

Theorembnj1340 32095 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 32096 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 32097* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theorembnj1351 32098* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorembnj1352 32099* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorembnj1361 32100* 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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