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Mirrors > Home > ILE Home > Th. List > 2ralbii | GIF version |
Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
ralbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2ralbii | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | ralbii 2441 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |
3 | 2 | ralbii 2441 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wral 2416 |
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-tru 1334 df-nf 1437 df-ral 2421 |
This theorem is referenced by: rmo4f 2882 ordsoexmid 4477 cnvsom 5082 fununi 5191 tpossym 6173 axpre-suploc 7710 isbasis2g 12212 |
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