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Mirrors > Home > ILE Home > Th. List > ralbi | GIF version |
Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.) |
Ref | Expression |
---|---|
ralbi | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 2521 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) | |
2 | rsp 2537 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))) | |
3 | 2 | imp 124 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
4 | 1, 3 | ralbida 2484 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 ∀wral 2468 |
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 1458 ax-gen 1460 ax-4 1521 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-ral 2473 |
This theorem is referenced by: uniiunlem 3259 iineq2 3918 ralrnmpt 5679 f1mpt 5793 mpo2eqb 6007 ralrnmpo 6012 cau3lem 11158 |
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