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Mirrors > Home > MPE Home > Th. List > rexanali | Structured version Visualization version GIF version |
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.) |
Ref | Expression |
---|---|
rexanali | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3202 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓)) | |
2 | iman 405 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) | |
3 | 2 | ralbii 3133 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓)) |
4 | 1, 3 | xchbinxr 338 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wral 3106 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-ral 3111 df-rex 3112 |
This theorem is referenced by: nrexralim 3225 wfi 6149 qsqueeze 12582 ncoprmgcdne1b 15984 elcls 21678 ist1-2 21952 haust1 21957 t1sep 21975 bwth 22015 1stccnp 22067 filufint 22525 fclscf 22630 pmltpc 24054 ovolgelb 24084 itg2seq 24346 radcnvlt1 25013 pntlem3 26193 umgr2edg1 27001 umgr2edgneu 27004 archiabl 30877 ordtconnlem1 31277 ceqsralv2 33069 frpoind 33193 frind 33198 nosupbnd1lem5 33325 limsucncmpi 33906 matunitlindflem1 35053 ftc1anclem5 35134 clsk3nimkb 40743 |
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