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Mirrors > Home > MPE Home > Th. List > r19.21v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.21v 1940. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) |
Ref | Expression |
---|---|
r19.21v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2.04 391 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) | |
2 | 1 | albii 1820 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) |
3 | 19.21v 1940 | . . 3 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) | |
4 | 2, 3 | bitri 277 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
5 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
6 | df-ral 3143 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
7 | 6 | imbi2i 338 | . 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
8 | 4, 5, 7 | 3bitr4i 305 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-ral 3143 |
This theorem is referenced by: r19.23v 3279 r19.32v 3340 rmo4 3721 2reu5lem3 3748 ra4v 3868 rmo3 3872 rmo3OLD 3873 dftr5 5175 reusv3 5306 tfinds2 7578 tfinds3 7579 wfr3g 7953 tfrlem1 8012 tfr3 8035 oeordi 8213 ordiso2 8979 ordtypelem7 8988 cantnf 9156 dfac12lem3 9571 ttukeylem5 9935 ttukeylem6 9936 fpwwe2lem8 10059 grudomon 10239 raluz2 12298 bpolycl 15406 ndvdssub 15760 gcdcllem1 15848 acsfn2 16934 pgpfac1 19202 pgpfac 19206 isdomn2 20072 islindf4 20982 isclo2 21696 1stccn 22071 kgencn 22164 txflf 22614 fclsopn 22622 nn0min 30536 bnj580 32185 bnj852 32193 rdgprc 33039 fpr3g 33122 conway 33264 filnetlem4 33729 poimirlem29 34936 heicant 34942 ntrneixb 40494 2rexrsb 43349 tfis2d 44832 |
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