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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 2349 | . 2 ⊢ Ⅎx∀x ∈ A (φ ↔ ψ) | |
2 | rsp 2363 | . . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → (x ∈ A → (φ ↔ ψ))) | |
3 | 2 | imp 115 | . 2 ⊢ ((∀x ∈ A (φ ↔ ψ) ∧ x ∈ A) → (φ ↔ ψ)) |
4 | 1, 3 | ralbida 2314 | 1 ⊢ (∀x ∈ A (φ ↔ ψ) → (∀x ∈ A φ ↔ ∀x ∈ A ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 ∀wral 2300 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-ral 2305 |
This theorem is referenced by: uniiunlem 3022 iineq2 3665 ralrnmpt 5252 f1mpt 5353 mpt22eqb 5552 ralrnmpt2 5557 |
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