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Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj534 32301* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → (∃𝑥𝜑𝜓))       (𝜒 → ∃𝑥(𝜑𝜓))
 
Theorembnj538 32302* 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 32303 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑀𝐷 → ∅ ∈ 𝑀)
 
Theorembnj551 32304 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 32305 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 32306 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 32307 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∃𝑥𝜒)
 
Theorembnj596 32308 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 32309* Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable condition on 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 32310 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜑)
 
Theorembnj643 32311 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜓)
 
Theorembnj645 32312 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → 𝜃)
 
Theorembnj658 32313 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜑𝜓𝜒))
 
Theorembnj667 32314 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorembnj705 32315 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj706 32316 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj707 32317 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj708 32318 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj721 32319 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒) → 𝜏)       ((𝜑𝜓𝜒𝜃) → 𝜏)
 
Theorembnj832 32320 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj835 32321 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj836 32322 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜓𝜏)       (𝜂𝜏)
 
Theorembnj837 32323 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒))    &   (𝜒𝜏)       (𝜂𝜏)
 
Theorembnj769 32324 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜑𝜏)       (𝜂𝜏)
 
Theorembnj770 32325 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜓𝜏)       (𝜂𝜏)
 
Theorembnj771 32326 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝜑𝜓𝜒𝜃))    &   (𝜒𝜏)       (𝜂𝜏)
 
Theorembnj887 32327 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜑′)    &   (𝜓𝜓′)    &   (𝜒𝜒′)    &   (𝜃𝜃′)       ((𝜑𝜓𝜒𝜃) ↔ (𝜑′𝜓′𝜒′𝜃′))
 
Theorembnj918 32328 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 32329* 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 32330 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑛𝐷𝑛 ∈ ω)
 
Theorembnj927 32331 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 32332 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 32333 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐵𝐴
 
Theorembnj937 32334* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)       (𝜑𝜓)
 
Theorembnj941 32335 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 32336 Technical lemma for bnj69 32573. 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 32337 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 32338 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜏𝜑)    &   (𝜏𝜓)    &   (𝜏𝜒)    &   (𝜏𝜃)       (𝜏 → (𝜑𝜓𝜒𝜃))
 
Theorembnj956 32339 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 32340* 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 32341 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 32342* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
 
Theorembnj1023 32343 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜓𝜒)       𝑥(𝜑𝜒)
 
Theorembnj1095 32344 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       (𝜑 → ∀𝑥𝜑)
 
Theorembnj1096 32345* 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 32346* 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 32347 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   (𝜒𝜑)       𝑥(𝜒𝜓)
 
Theorembnj1113 32348* 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 32349 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 32350 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   𝑥𝜑       𝜑
 
Theorembnj1138 32351 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶))
 
Theorembnj1142 32352 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 32353* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝐴 𝐵𝐵
 
Theorembnj1146 32354* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)        𝑥𝐴 𝐵𝐵
 
Theorembnj1149 32355 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 32356* 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 32357 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 32358 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 32359* 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 32360 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 32361 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜃𝑥𝐴)       (𝜃𝑥𝐵)
 
Theorembnj1212 32362* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}    &   (𝜃 ↔ (𝜒𝑥𝐵𝜏))       (𝜃𝑥𝐴)
 
Theorembnj1219 32363 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝜑𝜓𝜁))    &   (𝜃 ↔ (𝜒𝜏𝜂))       (𝜃𝜓)
 
Theorembnj1224 32364 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
¬ (𝜃𝜏𝜂)       ((𝜃𝜏) → ¬ 𝜂)
 
Theorembnj1230 32365* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}       (𝑦𝐵 → ∀𝑥 𝑦𝐵)
 
Theorembnj1232 32366 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜓)
 
Theorembnj1235 32367 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜒)
 
Theorembnj1239 32368 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜓𝜒) → ∃𝑥𝐴 𝜓)
 
Theorembnj1238 32369 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 32370 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝐴𝐵)    &   (𝜓𝐶 = 𝐴)       ((𝜑𝜓) → 𝐶𝐵)
 
Theorembnj1247 32371 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜃)
 
Theorembnj1254 32372 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒𝜃𝜏))       (𝜑𝜏)
 
Theorembnj1262 32373 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴𝐵    &   (𝜑𝐶 = 𝐴)       (𝜑𝐶𝐵)
 
Theorembnj1266 32374 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∃𝑥(𝜑𝜓))       (𝜒 → ∃𝑥𝜓)
 
Theorembnj1265 32375* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)       (𝜑𝜓)
 
Theorembnj1275 32376 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 32377 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 32378 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐵
 
Theorembnj1293 32379 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴𝐶
 
Theorembnj1294 32380 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝜓)
 
Theorembnj1299 32381 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 32382 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 32383* 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 32384* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 = {𝑥𝜑}       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theorembnj1322 32385 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
 
Theorembnj1340 32386 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 32387 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 32388* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
 
Theorembnj1351 32389* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theorembnj1352 32390* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theorembnj1361 32391* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theorembnj1366 32392* 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 32393* 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 32394* 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 32395* 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 32396* 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 32397 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑𝜓)
 
Theorembnj1400 32398* 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 32399* 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 32400 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)
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