![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > r19.26-2 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.26-2 1875. Version of r19.26 3112 with two quantifiers. (Contributed by NM, 10-Aug-2004.) |
Ref | Expression |
---|---|
r19.26-2 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3112 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) | |
2 | 1 | ralbii 3094 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) |
3 | r19.26 3112 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ral 3063 |
This theorem is referenced by: fununi 6620 tz7.48lem 8436 isffth2 17863 ispos2 18264 issgrpv 18608 issgrpn0 18609 isnsg2 19030 efgred 19609 dfrhm2 20242 df2idl2 20847 cpmatacl 22200 cpmatmcllem 22202 caucfil 24782 aalioulem6 25832 ajmoi 30089 adjmo 31063 prmidl2 32517 iccllysconn 34179 dfso3 34627 fvineqsnf1 36229 ispridl2 36844 ishlat2 38161 fiinfi 42257 ntrk1k3eqk13 42734 isrnghm 46624 |
Copyright terms: Public domain | W3C validator |