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Mirrors > Home > NFE Home > Th. List > r19.32v | GIF version |
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
r19.32v | ⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.21v 2701 | . 2 ⊢ (∀x ∈ A (¬ φ → ψ) ↔ (¬ φ → ∀x ∈ A ψ)) | |
2 | df-or 359 | . . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ)) | |
3 | 2 | ralbii 2638 | . 2 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ ∀x ∈ A (¬ φ → ψ)) |
4 | df-or 359 | . 2 ⊢ ((φ ∨ ∀x ∈ A ψ) ↔ (¬ φ → ∀x ∈ A ψ)) | |
5 | 1, 3, 4 | 3bitr4i 268 | 1 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∨ wo 357 ∀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 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-ral 2619 |
This theorem is referenced by: iinun2 4032 iinuni 4049 |
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