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Mirrors > Home > ILE Home > Th. List > rexralbidv | GIF version |
Description: Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) |
Ref | Expression |
---|---|
2ralbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexralbidv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | ralbidv 2438 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | rexbidv 2439 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wral 2417 ∃wrex 2418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-ral 2422 df-rex 2423 |
This theorem is referenced by: caucvgpr 7514 caucvgprpr 7544 caucvgsrlemgt1 7627 caucvgsrlemoffres 7632 axcaucvglemres 7731 cvg1nlemres 10789 rexfiuz 10793 resqrexlemgt0 10824 resqrexlemoverl 10825 resqrexlemglsq 10826 resqrexlemsqa 10828 resqrexlemex 10829 cau3lem 10918 caubnd2 10921 climi 11088 2clim 11102 ennnfonelemim 11973 lmcvg 12425 lmss 12454 txlm 12487 metcnpi 12723 metcnpi2 12724 elcncf 12768 cncfi 12773 limcimo 12842 cnplimclemr 12846 limccoap 12855 |
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