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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 3109 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 3109 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) | |
2 | 1 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) |
3 | r19.26 3109 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) | |
4 | 2, 3 | bitri 274 | 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 394 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ral 3060 |
This theorem is referenced by: fununi 6622 tz7.48lem 8443 isffth2 17871 ispos2 18272 issgrpv 18646 issgrpn0 18647 isnsg2 19072 efgred 19657 isrnghm 20332 dfrhm2 20365 df2idl2 21009 df2idl2rng 21037 cpmatacl 22438 cpmatmcllem 22440 caucfil 25031 aalioulem6 26086 ajmoi 30378 adjmo 31352 prmidl2 32833 iccllysconn 34539 dfso3 34993 fvineqsnf1 36594 ispridl2 37209 ishlat2 38526 fiinfi 42626 ntrk1k3eqk13 43103 |
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