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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 1901. Version of r19.23 3257 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 308 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)) | |
2 | 1 | ralbii 3115 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑)) |
3 | r19.21v 3125 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
4 | dfrex2 3186 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
5 | 4 | imbi1i 342 | . . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓)) |
6 | con1b 351 | . . 3 ⊢ ((¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
7 | 5, 6 | bitr2i 268 | . 2 ⊢ ((¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
8 | 2, 3, 7 | 3bitri 289 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wral 3088 ∃wrex 3089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-ral 3093 df-rex 3094 |
This theorem is referenced by: rexlimiv 3225 ralxpxfr2d 3554 uniiunlem 3953 2reu4lem 4349 dfiin2g 4828 iunss 4836 ralxfr2d 5165 ssrel2 5510 reliun 5540 idrefALT 5815 funimass4 6562 ralrnmpo 7107 kmlem12 9383 fimaxre3 11390 gcdcllem1 15711 vdwmc2 16174 iunocv 20530 islindf4 20687 ovolgelb 23787 dyadmax 23905 itg2leub 24041 mptelee 26387 nmoubi 28329 nmopub 29469 nmfnleub 29486 sigaclcu2 31024 untuni 32455 dfpo2 32511 elintfv 32527 heibor1lem 34529 ispsubsp2 36327 pmapglbx 36350 neik0pk1imk0 39760 2reuimp0 42720 |
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