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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 | r19.3rzv 4437 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | |
2 | 1 | imbi1d 344 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) |
3 | r19.35 3340 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl6rbbr 292 | 1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 ∅c0 4284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-dif 3932 df-nul 4285 |
This theorem is referenced by: ishlat3N 36523 hlsupr2 36556 |
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