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Mirrors > Home > MPE Home > Th. List > r19.23v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.23v 1943. Version of r19.23 3251 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) |
Ref | Expression |
---|---|
r19.23v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con34b 315 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)) | |
2 | 1 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑)) |
3 | r19.21v 3177 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
4 | dfrex2 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
5 | 4 | imbi1i 348 | . . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓)) |
6 | con1b 357 | . . 3 ⊢ ((¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
7 | 5, 6 | bitr2i 275 | . 2 ⊢ ((¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
8 | 2, 3, 7 | 3bitri 296 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wral 3059 ∃wrex 3068 |
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 1911 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rexlimivOLD 3182 ceqsralv 3512 ralxpxfr2d 3633 uniiunlem 4083 2reu4lem 4524 dfiin2g 5034 iunss 5047 ralxfr2d 5407 ssrel2 5784 reliun 5815 idrefALT 6111 dfpo2 6294 funimass4 6955 fnssintima 7361 ralrnmpo 7549 imaeqalov 7648 ttrclss 9717 kmlem12 10158 fimaxre3 12164 gcdcllem1 16444 vdwmc2 16916 iunocv 21453 islindf4 21612 ovolgelb 25229 dyadmax 25347 itg2leub 25484 eqscut2 27544 addsprop 27698 addsuniflem 27723 negsprop 27748 mulsprop 27825 mulsuniflem 27843 mptelee 28420 nmoubi 30292 nmopub 31428 nmfnleub 31445 sigaclcu2 33416 untuni 34982 elintfv 35040 heibor1lem 36980 ispsubsp2 38920 pmapglbx 38943 neik0pk1imk0 43100 2reuimp0 46120 |
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