P. 348 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34701-34800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bnj105 34701	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 1o ∈ V
Theorem	bnj115 34702	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 34703*	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 34704	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 34705	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 34706*	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 34707*	First-order logic and set theory. Revised to remove dependence on ax-reg 9604. (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 34708	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 34709	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 34710	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 34711*	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 34712	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 34713	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 34714	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 34715*	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 34716*	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 34717*	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 34718	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 34719	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 34720	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 34721	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 34722	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 34723	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 34724*	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 34725	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜑)
Theorem	bnj643 34726	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜓)
Theorem	bnj645 34727	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜃)
Theorem	bnj658 34728	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj667 34729	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	bnj705 34730	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj706 34731	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj707 34732	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj708 34733	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜃 → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj721 34734	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
Theorem	bnj832 34735	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj835 34736	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj836 34737	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj837 34738	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj769 34739	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj770 34740	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜓 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj771 34741	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))    &   ⊢ (𝜒 → 𝜏)    ⇒   ⊢ (𝜂 → 𝜏)
Theorem	bnj887 34742	∧-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 34743	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 34744*	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 34745	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 34746	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 34747	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 34748*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj941 34749	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 34750	Technical lemma for bnj69 34987. 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 34751	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 34752	∧-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 34753	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 34754*	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 34755	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 34756*	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 34757	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)
Theorem	bnj1095 34758	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	bnj1096 34759*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑))    ⇒   ⊢ (𝜓 → ∀𝑥𝜓)
Theorem	bnj1098 34760*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐷 = (ω ∖ {∅})    ⇒   ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
Theorem	bnj1101 34761	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 → 𝜓)
Theorem	bnj1113 34762*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)
Theorem	bnj1109 34763	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∃𝑥((𝐴 ≠ 𝐵 ∧ 𝜑) → 𝜓)    &   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
Theorem	bnj1131 34764	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	bnj1138 34765	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
Theorem	bnj1142 34766	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1143 34767*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵
Theorem	bnj1146 34768*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵
Theorem	bnj1149 34769	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bnj1185 34770*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
Theorem	bnj1196 34771	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	bnj1198 34772	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓′ ↔ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓′)
Theorem	bnj1209 34773*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))    ⇒   ⊢ (𝜒 → ∃𝑥𝜃)
Theorem	bnj1211 34774	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	bnj1213 34775	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜃 → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜃 → 𝑥 ∈ 𝐵)
Theorem	bnj1212 34776*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))    ⇒   ⊢ (𝜃 → 𝑥 ∈ 𝐴)
Theorem	bnj1219 34777	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜁))    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))    ⇒   ⊢ (𝜃 → 𝜓)
Theorem	bnj1224 34778	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ¬ (𝜃 ∧ 𝜏 ∧ 𝜂)    ⇒   ⊢ ((𝜃 ∧ 𝜏) → ¬ 𝜂)
Theorem	bnj1230 34779*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
Theorem	bnj1232 34780	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1235 34781	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	bnj1239 34782	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1238 34783	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1241 34784	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵)
Theorem	bnj1247 34785	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	bnj1254 34786	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	bnj1262 34787	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
Theorem	bnj1266 34788	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))    ⇒   ⊢ (𝜒 → ∃𝑥𝜓)
Theorem	bnj1265 34789*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1275 34790	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bnj1276 34791	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 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜃 → ∀𝑥𝜃)
Theorem	bnj1292 34792	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	bnj1293 34793	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	bnj1294 34794	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	bnj1299 34795	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	bnj1304 34796	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → ¬ 𝜒)    ⇒   ⊢ ¬ 𝜑
Theorem	bnj1316 34797*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	bnj1317 34798*	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	bnj1322 34799	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
Theorem	bnj1340 34800	First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∃𝑥𝜃)    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜓 → ∃𝑥𝜒)