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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 1868. (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 | bi2.04 376 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) | |
2 | 1 | albii 1747 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) |
3 | 19.21v 1868 | . . 3 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) | |
4 | 2, 3 | bitri 264 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
5 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
6 | df-ral 2917 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
7 | 6 | imbi2i 326 | . 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
8 | 4, 5, 7 | 3bitr4i 292 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 |
This theorem depends on definitions: df-bi 197 df-ex 1705 df-ral 2917 |
This theorem is referenced by: r19.23v 3023 r19.32v 3083 rmo4 3399 2reu5lem3 3415 ra4v 3524 rmo3 3528 dftr5 4755 reusv3 4876 tfinds2 7063 tfinds3 7064 wfr3g 7413 tfrlem1 7472 tfr3 7495 oeordi 7667 ordiso2 8420 ordtypelem7 8429 cantnf 8590 dfac12lem3 8967 ttukeylem5 9335 ttukeylem6 9336 fpwwe2lem8 9459 grudomon 9639 raluz2 11737 bpolycl 14783 ndvdssub 15133 gcdcllem1 15221 acsfn2 16324 pgpfac1 18479 pgpfac 18483 isdomn2 19299 islindf4 20177 isclo2 20892 1stccn 21266 kgencn 21359 txflf 21810 fclsopn 21818 nn0min 29567 bnj580 30983 bnj852 30991 rdgprc 31700 conway 31910 filnetlem4 32376 poimirlem29 33438 heicant 33444 ntrneixb 38393 2rexrsb 41171 tfis2d 42427 |
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