P. 352 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lpadlem2 35101	Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿)    ⇒   ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊)))
Theorem	lpadlen2 35102	Length of a left-padded word, in the case the given word 𝑊 is shorter than the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿)    ⇒   ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿)
Theorem	lpadmax 35103	Length of a left-padded word, in the general case, expressed with an if statement. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))
Theorem	lpadleft 35104	The contents of prefix of a left-padded word is always the letter 𝐶. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(𝐿 − (♯‘𝑊))))    ⇒   ⊢ (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘𝑁) = 𝐶)
Theorem	lpadright 35105	The suffix of a left-padded word the original word 𝑊. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 = if(𝐿 ≤ (♯‘𝑊), 0, (𝐿 − (♯‘𝑊))))    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘(𝑁 + 𝑀)) = (𝑊‘𝑁))
21.4  Mathbox for Jonathan Ben-Naim Note: On 4-Sep-2016 and after, 745 unused theorems were deleted from this mathbox, and 359 theorems used only once or twice were merged into their referencing theorems. The originals can be recovered from set.mm versions prior to this date.
Syntax	w-bnj17 35106	Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
Definition	df-bnj17 35107	Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
Syntax	c-bnj14 35108	Extend class notation with the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (New usage is discouraged.)
class pred(𝑋, 𝐴, 𝑅)
Definition	df-bnj14 35109*	Define the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
Syntax	w-bnj13 35110	Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴
Definition	df-bnj13 35111*	Define the following predicate: 𝑅 is set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
Syntax	w-bnj15 35112	Extend wff notation with the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 FrSe 𝐴
Definition	df-bnj15 35113	Define the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
Syntax	c-bnj18 35114	Extend class notation with the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. (New usage is discouraged.)
class trCl(𝑋, 𝐴, 𝑅)
Definition	df-bnj18 35115*	Define the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. This definition has been designed for facilitating verification that it is eliminable and that the $d restrictions are sound and complete. For a more readable definition see bnj882 35345. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
Syntax	w-bnj19 35116	Extend wff notation with the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (New usage is discouraged.)
wff TrFo(𝐵, 𝐴, 𝑅)
Definition	df-bnj19 35117*	Define the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
21.4.1  First-order logic and set theory
Theorem	bnj170 35118	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑))
Theorem	bnj240 35119	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → 𝜓′)    &   ⊢ (𝜒 → 𝜒′)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′))
Theorem	bnj248 35120	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃))
Theorem	bnj250 35121	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
Theorem	bnj251 35122	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))))
Theorem	bnj252 35123	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	bnj253 35124	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃))
Theorem	bnj255 35125	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)))
Theorem	bnj256 35126	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
Theorem	bnj257 35127	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒))
Theorem	bnj258 35128	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒))
Theorem	bnj268 35129	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓 ∧ 𝜃))
Theorem	bnj290 35130	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜃 ∧ 𝜓))
Theorem	bnj291 35131	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
Theorem	bnj312 35132	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃))
Theorem	bnj334 35133	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃))
Theorem	bnj345 35134	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj422 35135	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃 ∧ 𝜑 ∧ 𝜓))
Theorem	bnj432 35136	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜒 ∧ 𝜃) ∧ (𝜑 ∧ 𝜓)))
Theorem	bnj446 35137	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑))
Theorem	bnj23 35138*	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.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}    ⇒   ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑))
Theorem	bnj31 35139	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)
Theorem	bnj62 35140*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴)
Theorem	bnj89 35141*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝑍 ∈ V    ⇒   ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)
Theorem	bnj90 35142*	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.)
⊢ 𝑌 ∈ V    ⇒   ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)
Theorem	bnj101 35143	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ ∃𝑥𝜓
Theorem	bnj105 35144	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 1o ∈ V
Theorem	bnj115 35145	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ ∀𝑛 ∈ 𝐷 (𝜏 → 𝜃))    ⇒   ⊢ (𝜂 ↔ ∀𝑛((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))
Theorem	bnj132 35146*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
Theorem	bnj133 35147	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∃𝑥𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ∃𝑥𝜒)
Theorem	bnj156 35148	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))    &   ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)    &   ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)    &   ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)    ⇒   ⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
Theorem	bnj158 35149*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
Theorem	bnj168 35150*	First-order logic and set theory. Revised to remove dependence on ax-reg 9564. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚)
Theorem	bnj206 35151	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](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′))
Theorem	bnj216 35152	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)
Theorem	bnj219 35153	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)
Theorem	bnj226 35154*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
Theorem	bnj228 35155	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 ↔ ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)
Theorem	bnj519 35156	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⟩})
Theorem	bnj524 35157	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)
Theorem	bnj525 35158*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)
Theorem	bnj534 35159*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → (∃𝑥𝜑 ∧ 𝜓))    ⇒   ⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	bnj538 35160*	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]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)
Theorem	bnj529 35161	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ (𝑀 ∈ 𝐷 → ∅ ∈ 𝑀)
Theorem	bnj551 35162	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)
Theorem	bnj563 35163	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))    ⇒   ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)
Theorem	bnj564 35164	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))    ⇒   ⊢ (𝜏 → dom 𝑓 = 𝑚)
Theorem	bnj593 35165	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥𝜒)
Theorem	bnj596 35166	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	bnj610 35167*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′))    &   ⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
Theorem	bnj642 35168	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜑)
Theorem	bnj643 35169	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜓)
Theorem	bnj645 35170	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜃)
Theorem	bnj658 35171	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj667 35172	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	bnj705 35173	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj706 35174	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj707 35175	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj708 35176	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜃 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj721 35177	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj832 35178	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj835 35179	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj836 35180	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj837 35181	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj769 35182	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj770 35183	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj771 35184	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj887 35185	∧-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 35186	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 35187*	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 35188	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 35189	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 35190	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 35191*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj941 35192	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 35193	Technical lemma for bnj69 35429. 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 35194	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 35195	∧-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 35196	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 35197*	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 35198	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 35199*	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 35200	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)