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Mirrors > Home > MPE Home > Th. List > r19.45v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of one direction of 19.45 2094. The other direction holds when 𝐴 is nonempty, see r19.45zv 4020. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.45v | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.43 3074 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | id 22 | . . . 4 ⊢ (𝜑 → 𝜑) | |
3 | 2 | rexlimivw 3011 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜑) |
4 | 3 | orim1i 538 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
5 | 1, 4 | sylbi 206 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-ral 2901 df-rex 2902 |
This theorem is referenced by: (None) |
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