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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 1938. Version of r19.23 3244 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 3083 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑)) |
3 | r19.21v 3170 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
4 | dfrex2 3063 | . . . 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 3051 ∃wrex 3060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-ral 3052 df-rex 3061 |
This theorem is referenced by: rexlimivOLD 3175 ceqsralv 3504 ralxpxfr2d 3631 uniiunlem 4083 2reu4lem 4530 dfiin2g 5042 iunss 5055 ralxfr2d 5416 ssrel2 5793 reliun 5824 idrefALT 6125 dfpo2 6309 funimass4 6969 fnssintima 7376 ralrnmpo 7567 imaeqalov 7667 ttrclss 9765 kmlem12 10206 fimaxre3 12214 gcdcllem1 16501 vdwmc2 16983 iunocv 21679 islindf4 21838 ovolgelb 25503 dyadmax 25621 itg2leub 25758 eqscut2 27839 addsprop 27993 addsuniflem 28018 negsprop 28047 mulsprop 28134 mulsuniflem 28153 mptelee 28832 nmoubi 30708 nmopub 31844 nmfnleub 31861 sigaclcu2 33955 untuni 35533 elintfv 35590 heibor1lem 37512 ispsubsp2 39447 pmapglbx 39470 neik0pk1imk0 43732 2reuimp0 46745 |
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