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Mirrors > Home > NFE Home > Th. List > r19.26m | GIF version |
Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
r19.26m | ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x ∈ A φ ∧ ∀x ∈ B ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1593 | . 2 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ))) | |
2 | df-ral 2619 | . . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
3 | df-ral 2619 | . . 3 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ)) | |
4 | 2, 3 | anbi12i 678 | . 2 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ B ψ) ↔ (∀x(x ∈ A → φ) ∧ ∀x(x ∈ B → ψ))) |
5 | 1, 4 | bitr4i 243 | 1 ⊢ (∀x((x ∈ A → φ) ∧ (x ∈ B → ψ)) ↔ (∀x ∈ A φ ∧ ∀x ∈ B ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ral 2619 |
This theorem is referenced by: (None) |
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