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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 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓)) | |
2 | iman 401 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) | |
3 | 2 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓)) |
4 | 1, 3 | xchbinxr 335 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wral 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-ral 3060 df-rex 3069 |
This theorem is referenced by: nrexralim 3135 ceqsralbv 3657 frpoind 6365 wfiOLD 6374 frind 9788 qsqueeze 13240 ncoprmgcdne1b 16684 elcls 23097 ist1-2 23371 haust1 23376 t1sep 23394 bwth 23434 1stccnp 23486 filufint 23944 fclscf 24049 pmltpc 25499 ovolgelb 25529 itg2seq 25792 radcnvlt1 26476 pntlem3 27668 nosupbnd1lem5 27772 noinfbnd1lem5 27787 umgr2edg1 29243 umgr2edgneu 29246 archiabl 33188 ordtconnlem1 33885 limsucncmpi 36428 matunitlindflem1 37603 ftc1anclem5 37684 clsk3nimkb 44030 |
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