Theorem List for Intuitionistic Logic Explorer - 2301-2400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | necon2bi 2301 |
Contrapositive inference for inequality. (Contributed by NM,
1-Apr-2007.)
|
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
|
Theorem | necon2i 2302 |
Contrapositive inference for inequality. (Contributed by NM,
18-Mar-2007.)
|
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) |
|
Theorem | necon2ad 2303 |
Contrapositive inference for inequality. (Contributed by NM,
19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
|
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
|
Theorem | necon2bd 2304 |
Contrapositive inference for inequality. (Contributed by NM,
13-Apr-2007.)
|
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
|
Theorem | necon2d 2305 |
Contrapositive inference for inequality. (Contributed by NM,
28-Dec-2008.)
|
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) |
|
Theorem | necon1abiidc 2306 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵)) ⇒ ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑)) |
|
Theorem | necon1bbiidc 2307 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜑)) ⇒ ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 ↔ 𝐴 = 𝐵)) |
|
Theorem | necon1abiddc 2308 |
Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓))) |
|
Theorem | necon1bbiddc 2309 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵))) |
|
Theorem | necon2abiidc 2310 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑)) ⇒ ⊢ (DECID 𝜑 → (𝜑 ↔ 𝐴 ≠ 𝐵)) |
|
Theorem | necon2bbii 2311 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝐴 = 𝐵 → (𝜑 ↔ 𝐴 ≠ 𝐵)) ⇒ ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑)) |
|
Theorem | necon2abiddc 2312 |
Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵))) |
|
Theorem | necon2bbiddc 2313 |
Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓))) |
|
Theorem | necon4aidc 2314 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑)) ⇒ ⊢ (DECID 𝐴 = 𝐵 → (𝜑 → 𝐴 = 𝐵)) |
|
Theorem | necon4idc 2315 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
16-May-2018.)
|
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) |
|
Theorem | necon4addc 2316 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
17-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵))) |
|
Theorem | necon4bddc 2317 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
17-May-2018.)
|
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓))) |
|
Theorem | necon4ddc 2318 |
Contrapositive inference for inequality. (Contributed by Jim Kingdon,
17-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵))) |
|
Theorem | necon4abiddc 2319 |
Contrapositive law deduction for inequality. (Contributed by Jim
Kingdon, 18-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓)))) |
|
Theorem | necon4bbiddc 2320 |
Contrapositive law deduction for inequality. (Contributed by Jim
Kingdon, 19-May-2018.)
|
⊢ (𝜑 → (DECID 𝜓 → (DECID
𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (DECID
𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵)))) |
|
Theorem | necon4biddc 2321 |
Contrapositive law deduction for inequality. (Contributed by Jim
Kingdon, 19-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)))) |
|
Theorem | necon1addc 2322 |
Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
19-May-2018.)
|
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓))) |
|
Theorem | necon1bddc 2323 |
Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
19-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵))) |
|
Theorem | necon1ddc 2324 |
Contrapositive law deduction for inequality. (Contributed by Jim
Kingdon, 19-May-2018.)
|
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷))) ⇒ ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))) |
|
Theorem | neneqad 2325 |
If it is not the case that two classes are equal, they are unequal.
Converse of neneqd 2267. One-way deduction form of df-ne 2247.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
|
Theorem | nebidc 2326 |
Contraposition law for inequality. (Contributed by Jim Kingdon,
19-May-2018.)
|
⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))) |
|
Theorem | pm13.18 2327 |
Theorem *13.18 in [WhiteheadRussell]
p. 178. (Contributed by Andrew
Salmon, 3-Jun-2011.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
|
Theorem | pm13.181 2328 |
Theorem *13.181 in [WhiteheadRussell]
p. 178. (Contributed by Andrew
Salmon, 3-Jun-2011.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
|
Theorem | pm2.21ddne 2329 |
A contradiction implies anything. Equality/inequality deduction form.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝜓) |
|
Theorem | necom 2330 |
Commutation of inequality. (Contributed by NM, 14-May-1999.)
|
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) |
|
Theorem | necomi 2331 |
Inference from commutative law for inequality. (Contributed by NM,
17-Oct-2012.)
|
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 |
|
Theorem | necomd 2332 |
Deduction from commutative law for inequality. (Contributed by NM,
12-Feb-2008.)
|
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
|
Theorem | neanior 2333 |
A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
|
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷)) |
|
Theorem | ne3anior 2334 |
A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
(Proof rewritten by Jim Kingdon, 19-May-2018.)
|
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹)) |
|
Theorem | nemtbir 2335 |
An inference from an inequality, related to modus tollens. (Contributed
by NM, 13-Apr-2007.)
|
⊢ 𝐴 ≠ 𝐵
& ⊢ (𝜑 ↔ 𝐴 = 𝐵) ⇒ ⊢ ¬ 𝜑 |
|
Theorem | nelne1 2336 |
Two classes are different if they don't contain the same element.
(Contributed by NM, 3-Feb-2012.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶) |
|
Theorem | nelne2 2337 |
Two classes are different if they don't belong to the same class.
(Contributed by NM, 25-Jun-2012.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) |
|
Theorem | nfne 2338 |
Bound-variable hypothesis builder for inequality. (Contributed by NM,
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ≠ 𝐵 |
|
Theorem | nfned 2339 |
Bound-variable hypothesis builder for inequality. (Contributed by NM,
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
|
2.1.4.2 Negated membership
|
|
Syntax | wnel 2340 |
Extend wff notation to include negated membership.
|
wff 𝐴 ∉ 𝐵 |
|
Definition | df-nel 2341 |
Define negated membership. (Contributed by NM, 7-Aug-1994.)
|
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) |
|
Theorem | neli 2342 |
Inference associated with df-nel 2341. (Contributed by BJ,
7-Jul-2018.)
|
⊢ 𝐴 ∉ 𝐵 ⇒ ⊢ ¬ 𝐴 ∈ 𝐵 |
|
Theorem | nelir 2343 |
Inference associated with df-nel 2341. (Contributed by BJ,
7-Jul-2018.)
|
⊢ ¬ 𝐴 ∈ 𝐵 ⇒ ⊢ 𝐴 ∉ 𝐵 |
|
Theorem | neleq1 2344 |
Equality theorem for negated membership. (Contributed by NM,
20-Nov-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
|
Theorem | neleq2 2345 |
Equality theorem for negated membership. (Contributed by NM,
20-Nov-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) |
|
Theorem | neleq12d 2346 |
Equality theorem for negated membership. (Contributed by FL,
10-Aug-2016.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
|
Theorem | nfnel 2347 |
Bound-variable hypothesis builder for negated membership. (Contributed
by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro,
7-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵 |
|
Theorem | nfneld 2348 |
Bound-variable hypothesis builder for negated membership. (Contributed
by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro,
7-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵) |
|
2.1.5 Restricted quantification
|
|
Syntax | wral 2349 |
Extend wff notation to include restricted universal quantification.
|
wff ∀𝑥 ∈ 𝐴 𝜑 |
|
Syntax | wrex 2350 |
Extend wff notation to include restricted existential quantification.
|
wff ∃𝑥 ∈ 𝐴 𝜑 |
|
Syntax | wreu 2351 |
Extend wff notation to include restricted existential uniqueness.
|
wff ∃!𝑥 ∈ 𝐴 𝜑 |
|
Syntax | wrmo 2352 |
Extend wff notation to include restricted "at most one."
|
wff ∃*𝑥 ∈ 𝐴 𝜑 |
|
Syntax | crab 2353 |
Extend class notation to include the restricted class abstraction (class
builder).
|
class {𝑥 ∈ 𝐴 ∣ 𝜑} |
|
Definition | df-ral 2354 |
Define restricted universal quantification. Special case of Definition
4.15(3) of [TakeutiZaring] p. 22.
(Contributed by NM, 19-Aug-1993.)
|
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
|
Definition | df-rex 2355 |
Define restricted existential quantification. Special case of Definition
4.15(4) of [TakeutiZaring] p. 22.
(Contributed by NM, 30-Aug-1993.)
|
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
|
Definition | df-reu 2356 |
Define restricted existential uniqueness. (Contributed by NM,
22-Nov-1994.)
|
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
|
Definition | df-rmo 2357 |
Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
|
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
|
Definition | df-rab 2358 |
Define a restricted class abstraction (class builder), which is the class
of all 𝑥 in 𝐴 such that 𝜑 is true. Definition of
[TakeutiZaring] p. 20. (Contributed
by NM, 22-Nov-1994.)
|
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
|
Theorem | ralnex 2359 |
Relationship between restricted universal and existential quantifiers.
(Contributed by NM, 21-Jan-1997.)
|
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
|
Theorem | rexnalim 2360 |
Relationship between restricted universal and existential quantifiers. In
classical logic this would be a biconditional. (Contributed by Jim
Kingdon, 17-Aug-2018.)
|
⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | ralexim 2361 |
Relationship between restricted universal and existential quantifiers.
(Contributed by Jim Kingdon, 17-Aug-2018.)
|
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
|
Theorem | rexalim 2362 |
Relationship between restricted universal and existential quantifiers.
(Contributed by Jim Kingdon, 17-Aug-2018.)
|
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) |
|
Theorem | ralbida 2363 |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 6-Oct-2003.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rexbida 2364 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 6-Oct-2003.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | ralbidva 2365* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 4-Mar-1997.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rexbidva 2366* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 9-Mar-1997.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | ralbid 2367 |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 27-Jun-1998.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rexbid 2368 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 27-Jun-1998.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | ralbidv 2369* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 20-Nov-1994.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | rexbidv 2370* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 20-Nov-1994.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
|
Theorem | ralbidv2 2371* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 6-Apr-1997.)
|
⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
|
Theorem | rexbidv2 2372* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 22-May-1999.)
|
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
|
Theorem | ralbii 2373 |
Inference adding restricted universal quantifier to both sides of an
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
Carneiro, 17-Oct-2016.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) |
|
Theorem | rexbii 2374 |
Inference adding restricted existential quantifier to both sides of an
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
Carneiro, 17-Oct-2016.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓) |
|
Theorem | 2ralbii 2375 |
Inference adding two restricted universal quantifiers to both sides of
an equivalence. (Contributed by NM, 1-Aug-2004.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
|
Theorem | 2rexbii 2376 |
Inference adding two restricted existential quantifiers to both sides of
an equivalence. (Contributed by NM, 11-Nov-1995.)
|
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
|
Theorem | ralbii2 2377 |
Inference adding different restricted universal quantifiers to each side
of an equivalence. (Contributed by NM, 15-Aug-2005.)
|
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓) |
|
Theorem | rexbii2 2378 |
Inference adding different restricted existential quantifiers to each
side of an equivalence. (Contributed by NM, 4-Feb-2004.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓) |
|
Theorem | raleqbii 2379 |
Equality deduction for restricted universal quantifier, changing both
formula and quantifier domain. Inference form. (Contributed by David
Moews, 1-May-2017.)
|
⊢ 𝐴 = 𝐵
& ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) |
|
Theorem | rexeqbii 2380 |
Equality deduction for restricted existential quantifier, changing both
formula and quantifier domain. Inference form. (Contributed by David
Moews, 1-May-2017.)
|
⊢ 𝐴 = 𝐵
& ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
|
Theorem | ralbiia 2381 |
Inference adding restricted universal quantifier to both sides of an
equivalence. (Contributed by NM, 26-Nov-2000.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) |
|
Theorem | rexbiia 2382 |
Inference adding restricted existential quantifier to both sides of an
equivalence. (Contributed by NM, 26-Oct-1999.)
|
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓) |
|
Theorem | 2rexbiia 2383* |
Inference adding two restricted existential quantifiers to both sides of
an equivalence. (Contributed by NM, 1-Aug-2004.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
|
Theorem | r2alf 2384* |
Double restricted universal quantification. (Contributed by Mario
Carneiro, 14-Oct-2016.)
|
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
|
Theorem | r2exf 2385* |
Double restricted existential quantification. (Contributed by Mario
Carneiro, 14-Oct-2016.)
|
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
|
Theorem | r2al 2386* |
Double restricted universal quantification. (Contributed by NM,
19-Nov-1995.)
|
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
|
Theorem | r2ex 2387* |
Double restricted existential quantification. (Contributed by NM,
11-Nov-1995.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
|
Theorem | 2ralbida 2388* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 24-Feb-2004.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑦𝜑
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | 2ralbidva 2389* |
Formula-building rule for restricted universal quantifiers (deduction
rule). (Contributed by NM, 4-Mar-1997.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | 2rexbidva 2390* |
Formula-building rule for restricted existential quantifiers (deduction
rule). (Contributed by NM, 15-Dec-2004.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | 2ralbidv 2391* |
Formula-building rule for restricted universal quantifiers (deduction
rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon
Jaroszewicz, 16-Mar-2007.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | 2rexbidv 2392* |
Formula-building rule for restricted existential quantifiers (deduction
rule). (Contributed by NM, 28-Jan-2006.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | rexralbidv 2393* |
Formula-building rule for restricted quantifiers (deduction rule).
(Contributed by NM, 28-Jan-2006.)
|
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
|
Theorem | ralinexa 2394 |
A transformation of restricted quantifiers and logical connectives.
(Contributed by NM, 4-Sep-2005.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
|
Theorem | risset 2395* |
Two ways to say "𝐴 belongs to 𝐵." (Contributed by
NM,
22-Nov-1994.)
|
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) |
|
Theorem | hbral 2396 |
Bound-variable hypothesis builder for restricted quantification.
(Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy,
13-Dec-2009.)
|
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
& ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑) |
|
Theorem | hbra1 2397 |
𝑥
is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM,
18-Oct-1996.)
|
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | nfra1 2398 |
𝑥
is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.)
(Revised by Mario Carneiro, 7-Oct-2016.)
|
⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 |
|
Theorem | nfraldxy 2399* |
Not-free for restricted universal quantification where 𝑥 and 𝑦
are distinct. See nfraldya 2401 for a version with 𝑦 and
𝐴
distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
|
Theorem | nfrexdxy 2400* |
Not-free for restricted existential quantification where 𝑥 and 𝑦
are distinct. See nfrexdya 2402 for a version with 𝑦 and
𝐴
distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
|
⊢ Ⅎ𝑦𝜑
& ⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |