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Mirrors > Home > NFE Home > Th. List > r19.43 | GIF version |
Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.43 | ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.35 2758 | . 2 ⊢ (∃x ∈ A (¬ φ → ψ) ↔ (∀x ∈ A ¬ φ → ∃x ∈ A ψ)) | |
2 | df-or 359 | . . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ)) | |
3 | 2 | rexbii 2639 | . 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ ∃x ∈ A (¬ φ → ψ)) |
4 | df-or 359 | . . 3 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (¬ ∃x ∈ A φ → ∃x ∈ A ψ)) | |
5 | ralnex 2624 | . . . 4 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ) | |
6 | 5 | imbi1i 315 | . . 3 ⊢ ((∀x ∈ A ¬ φ → ∃x ∈ A ψ) ↔ (¬ ∃x ∈ A φ → ∃x ∈ A ψ)) |
7 | 4, 6 | bitr4i 243 | . 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A ¬ φ → ∃x ∈ A ψ)) |
8 | 1, 3, 7 | 3bitr4i 268 | 1 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∨ 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: r19.44av 2767 r19.45av 2768 r19.45zv 3647 iunun 4046 nncdiv3 6277 |
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