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Mirrors > Home > MPE Home > Th. List > r19.26-2 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.26-2 1872. Version of r19.26 3170 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 3170 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) | |
2 | 1 | ralbii 3165 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) |
3 | r19.26 3170 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) | |
4 | 2, 3 | bitri 277 | 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 208 ∧ wa 398 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ral 3143 |
This theorem is referenced by: fununi 6429 tz7.48lem 8077 isffth2 17186 ispos2 17558 issgrpv 17903 issgrpn0 17904 isnsg2 18308 efgred 18874 dfrhm2 19469 cpmatacl 21324 cpmatmcllem 21326 caucfil 23886 aalioulem6 24926 ajmoi 28635 adjmo 29609 prmidl2 30958 iccllysconn 32497 dfso3 32950 fvineqsnf1 34694 ispridl2 35331 ishlat2 36504 fiinfi 39952 ntrk1k3eqk13 40420 isrnghm 44183 |
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