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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 2483 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |
3 | 2 | ralbii 2483 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-ral 2460 |
This theorem is referenced by: rmo4f 2935 ordsoexmid 4557 cnvsom 5167 fununi 5279 tpossym 6270 axpre-suploc 7879 issubm 12740 isbasis2g 13176 |
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