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| Mirrors > Home > MPE Home > Th. List > r19.37zv | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.) | 
| Ref | Expression | 
|---|---|
| r19.37zv | ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r19.35 3107 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
| 2 | r19.3rzv 4498 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | |
| 3 | 2 | imbi1d 341 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) | 
| 4 | 1, 3 | bitr4id 290 | 1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-ne 2940 df-ral 3061 df-rex 3070 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: ishlat3N 39356 hlsupr2 39390 | 
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