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Mirrors > Home > ILE Home > Th. List > 2ralbidv | GIF version |
Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.) |
Ref | Expression |
---|---|
2ralbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
2ralbidv | ⊢ (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 2435 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | ralbidv 2435 | 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 2414 |
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 1423 ax-gen 1425 ax-4 1487 ax-17 1506 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-ral 2419 |
This theorem is referenced by: cbvral3v 2662 poeq1 4216 soeq1 4232 isoeq1 5695 isoeq2 5696 isoeq3 5697 fnmpoovd 6105 smoeq 6180 xpf1o 6731 elinp 7275 cauappcvgpr 7463 seq3caopr2 10248 addcn2 11072 mulcn2 11074 ispsmet 12481 ismet 12502 isxmet 12503 addcncntoplem 12709 elcncf 12718 |
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