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Mirrors > Home > NFE Home > Th. List > r19.30 | GIF version |
Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.) |
Ref | Expression |
---|---|
r19.30 | ⊢ (∀x ∈ A (φ ∨ ψ) → (∀x ∈ A φ ∨ ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralim 2685 | . 2 ⊢ (∀x ∈ A (¬ ψ → φ) → (∀x ∈ A ¬ ψ → ∀x ∈ A φ)) | |
2 | orcom 376 | . . . 4 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ)) | |
3 | df-or 359 | . . . 4 ⊢ ((ψ ∨ φ) ↔ (¬ ψ → φ)) | |
4 | 2, 3 | bitri 240 | . . 3 ⊢ ((φ ∨ ψ) ↔ (¬ ψ → φ)) |
5 | 4 | ralbii 2638 | . 2 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ ∀x ∈ A (¬ ψ → φ)) |
6 | orcom 376 | . . 3 ⊢ ((∀x ∈ A φ ∨ ¬ ∀x ∈ A ¬ ψ) ↔ (¬ ∀x ∈ A ¬ ψ ∨ ∀x ∈ A φ)) | |
7 | dfrex2 2627 | . . . 4 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ) | |
8 | 7 | orbi2i 505 | . . 3 ⊢ ((∀x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A φ ∨ ¬ ∀x ∈ A ¬ ψ)) |
9 | imor 401 | . . 3 ⊢ ((∀x ∈ A ¬ ψ → ∀x ∈ A φ) ↔ (¬ ∀x ∈ A ¬ ψ ∨ ∀x ∈ A φ)) | |
10 | 6, 8, 9 | 3bitr4i 268 | . 2 ⊢ ((∀x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A ¬ ψ → ∀x ∈ A φ)) |
11 | 1, 5, 10 | 3imtr4i 257 | 1 ⊢ (∀x ∈ A (φ ∨ ψ) → (∀x ∈ A φ ∨ ∃x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 357 ∀wral 2614 ∃wrex 2615 |
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-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 df-rex 2620 |
This theorem is referenced by: (None) |
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