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Mirrors > Home > NFE Home > Th. List > r19.44av | GIF version |
Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.44av | ⊢ (∃x ∈ A (φ ∨ ψ) → (∃x ∈ A φ ∨ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.43 2766 | . 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) | |
2 | idd 21 | . . . 4 ⊢ (x ∈ A → (ψ → ψ)) | |
3 | 2 | rexlimiv 2732 | . . 3 ⊢ (∃x ∈ A ψ → ψ) |
4 | 3 | orim2i 504 | . 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) → (∃x ∈ A φ ∨ ψ)) |
5 | 1, 4 | sylbi 187 | 1 ⊢ (∃x ∈ A (φ ∨ ψ) → (∃x ∈ A φ ∨ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 357 ∈ wcel 1710 ∃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-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 df-rex 2620 |
This theorem is referenced by: (None) |
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