P. 452 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45101-45200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm10.14 45101	Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	pm10.251 45102	Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	pm10.252 45103	Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
Theorem	pm10.253 45104	Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)
Theorem	albitr 45105	Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → ∀𝑥(𝜑 ↔ 𝜒))
Theorem	pm10.42 45106	Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∨ 𝜓))
Theorem	pm10.52 45107*	Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))
Theorem	pm10.53 45108	Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
Theorem	pm10.541 45109*	Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.542 45110*	Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.55 45111	Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥(𝜑 → 𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.56 45112	Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒))
Theorem	pm10.57 45113	Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒)))
21.41.2  Principia Mathematica * 11
Theorem	2alanimi 45114	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦𝜒)
Theorem	2al2imi 45115	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	pm11.11 45116	Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ 𝜑    ⇒   ⊢ ∀𝑧∀𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
Theorem	pm11.12 45117*	Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))
Theorem	19.21vv 45118*	Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1972. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))
Theorem	2alim 45119	Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	2albi 45120	Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	2exim 45121	Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	2exbi 45122	Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	spsbce-2 45123	Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	19.33-2 45124	Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))
Theorem	19.36vv 45125*	Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))
Theorem	19.31vv 45126*	Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))
Theorem	19.37vv 45127*	Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))
Theorem	19.28vv 45128*	Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∀𝑥∀𝑦𝜑))
Theorem	pm11.52 45129	Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓))
Theorem	aaanv 45130*	Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2368. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))
Theorem	pm11.57 45131*	Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.58 45132*	Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.59 45133*	Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
Theorem	pm11.6 45134*	Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	pm11.61 45135*	Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
Theorem	pm11.62 45136*	Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒)))
Theorem	pm11.63 45137	Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))
Theorem	pm11.7 45138	Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
Theorem	pm11.71 45139*	Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))))
21.41.3  Predicate Calculus
Theorem	sbeqal1 45140*	If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)
Theorem	sbeqal1i 45141*	Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑦 = 𝑧
Theorem	sbeqal2i 45142*	If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑧 = 𝑦
Theorem	axc5c4c711 45143	Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1828 as the inference rule. This proof extends the idea of axc5c711 39724 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
Theorem	axc5c4c711toc5 45144	Rederivation of sp 2222 from axc5c4c711 45143. Note that ax6 2419 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 2001 instead of ax6 2419, so that this rederivation requires only ax6v 2001 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axc5c4c711toc4 45145	Rederivation of axc4 2357 from axc5c4c711 45143. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	axc5c4c711toc7 45146	Rederivation of axc7 2353 from axc5c4c711 45143. Note that neither axc7 2353 nor ax-11 2195 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc5c4c711to11 45147	Rederivation of ax-11 2195 from axc5c4c711 45143. Note that ax-11 2195 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	axc11next 45148*	This theorem shows that, given axextb 2741, we can derive a version of axc11n 2461. However, it is weaker than axc11n 2461 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
21.41.4  Principia Mathematica * 13 and * 14
Theorem	pm13.13a 45149	One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)
Theorem	pm13.13b 45150	Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑)
Theorem	pm13.14 45151	Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)
Theorem	pm13.192 45152*	Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Theorem	pm13.193 45153	Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦))
Theorem	pm13.194 45154	Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))
Theorem	pm13.195 45155*	Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3775. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Theorem	pm13.196a 45156*	Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥))
Theorem	2sbc6g 45157*	Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Theorem	2sbc5g 45158*	Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Theorem	iotain 45159	Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
Theorem	iotaexeu 45160	The iota class exists. This theorem does not require ax-nul 5272 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
Theorem	iotasbc 45161*	Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define ℩ in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
Theorem	iotasbc2 45162*	Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)))
Theorem	pm14.12 45163*	Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	pm14.122a 45164*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
Theorem	pm14.122b 45165*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Theorem	pm14.122c 45166*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Theorem	pm14.123a 45167*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
Theorem	pm14.123b 45168*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))
Theorem	pm14.123c 45169*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))
Theorem	pm14.18 45170	Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
Theorem	iotaequ 45171*	Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (℩𝑥𝑥 = 𝑦) = 𝑦
Theorem	iotavalb 45172*	Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6514. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
Theorem	iotasbc5 45173*	Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)))
Theorem	pm14.24 45174*	Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑)))
Theorem	iotavalsb 45175*	Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))
Theorem	sbiota1 45176	Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Theorem	sbaniota 45177	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Theorem	iotasbcq 45178	Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([(℩𝑥𝜑) / 𝑦]𝜒 ↔ [(℩𝑥𝜓) / 𝑦]𝜒))
21.41.5  Set Theory
Theorem	elnev 45179*	Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Theorem	rusbcALT 45180	A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
Theorem	compeq 45181*	Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}
Theorem	compne 45182	The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.)
⊢ (V ∖ 𝐴) ≠ 𝐴
Theorem	compab 45183	Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}
Theorem	conss2 45184	Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))
Theorem	conss1 45185	Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴)
Theorem	ralbidar 45186	More general form of ralbida 3279. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexbidar 45187	More general form of rexbida 3280. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	dropab1 45188	Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
Theorem	dropab2 45189	Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})
Theorem	ipo0 45190	If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)
Theorem	ifr0 45191	A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)
Theorem	fvsb 45192*	Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
Theorem	fveqsb 45193*	Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
Theorem	xpexb 45194	A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)
Theorem	trelpss 45195	An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5647, ax-reg 9556 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)
21.41.6  Arithmetic
Theorem	addcomgi 45196	Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)
21.41.7  Geometry
Syntax	cplusr 45197	Introduce the operation of vector addition.
class +𝑟
Syntax	cminusr 45198	Introduce the operation of vector subtraction.
class -𝑟
Syntax	ctimesr 45199	Introduce the operation of scalar multiplication.
class .𝑣
Syntax	cptdfc 45200	PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
class PtDf(𝐴, 𝐵)