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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 3255 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | r19.3rzv 4410 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | |
3 | 2 | imbi1d 345 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) |
4 | 1, 3 | bitr4id 293 | 1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ∅c0 4237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-ne 2941 df-ral 3066 df-rex 3067 df-dif 3869 df-nul 4238 |
This theorem is referenced by: ishlat3N 37105 hlsupr2 37138 |
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